Sleeping Beauty and De Nunc Updating

University of Massachusetts Amherst University of Massachusetts Amherst

ScholarWorks@UMass Amherst ScholarWorks@UMass Amherst

Open Access Dissertations

5-2010

Sleeping Beauty and De Nunc Updating Sleeping Beauty and De Nunc Updating

Namjoong Kim University of Massachusetts Amherst

Follow this and additional works at: https://scholarworks.umass.edu/open_access_dissertations

Part of the Philosophy Commons

Recommended Citation Recommended Citation Kim, Namjoong, "Sleeping Beauty and De Nunc Updating" (2010). Open Access Dissertations. 242. https://scholarworks.umass.edu/open_access_dissertations/242

This Open Access Dissertation is brought to you for free and open access by ScholarWorks@UMass Amherst. It has been accepted for inclusion in Open Access Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected].

https://scholarworks.umass.edu/

https://scholarworks.umass.edu/open_access_dissertations

https://scholarworks.umass.edu/open_access_dissertations?utm_source=scholarworks.umass.edu%2Fopen_access_dissertations%2F242&utm_medium=PDF&utm_campaign=PDFCoverPages

http://network.bepress.com/hgg/discipline/525?utm_source=scholarworks.umass.edu%2Fopen_access_dissertations%2F242&utm_medium=PDF&utm_campaign=PDFCoverPages

https://scholarworks.umass.edu/open_access_dissertations/242?utm_source=scholarworks.umass.edu%2Fopen_access_dissertations%2F242&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

SLEEPING BEAUTY AND DE NUNC UPDATING

A Dissertation Presented

by

NAMJOONG KIM

Submitted to the Graduate School of the

University of Massachusetts Amherst in partial fulfillment

of the requirements for the degree of

DOCTOR OF PHILOSOPHY

May 2010

Philosophy


A Dissertation Presented

by

NAMJOONG KIM

Approved as to style and content by:

Phillip Bricker, Chair

Hillary Kornblith, Member

Christopher Meacham, Member

Lyn Frazier, Member

Phillip Bricker, Department Head

Philosophy

iv

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor, Phillip Bricker. Ever since I

came up with a small idea about how to update one’s degrees of self-locating beliefs, he

patiently helped me to develop it into a sophisticated theory as presented in this

dissertation. During that process, he continued to offer lots of encouragement, advice, and

sometimes criticisms. Truly, this dissertation would not have been born without his help.

I am greatly indebted to the other members of the committee as well. Through

his teaching and writing, Hillary Kornblith taught me how to philosophize basic

epistemological problems in a clear, effective way. Thanks to him, I better understood the

connection between many issues of mainstream and formal epistemologies. In particular,

he helped me identify the conditions that ought to be satisfied by any cogent rule for

updating.

I cannot overemphasize the important role Chris Meacham played in the early

development of my thoughts. Only after I read his “Sleeping Beauty and the Dynamics of

De Se Beliefs,” did I realize that a new rule for updating the degrees of self-locating

beliefs was essential for the right solution of the Sleeping Beauty problem. After he came

to UMass, we enjoyed a good number of discussions about this dissertation and many

other issues in the philosophy of probability.

It has been a privilege to have Lyn Frazier on my committee. She gave me

invaluable opportunities to think about philosophical probabilism from an outsider’s

point of view. Although she is a linguist and I am a philosopher, our differences worked

as a catalyst of new ideas rather than a barrier between disciplines.

v

The second chapter of this dissertation was published in Synthese. So its

copyright belongs to Springer-Verlag New York, LLC. I am grateful to the editor of

Synthese, Wiebe van der Hoek, for allowing me to include it as a chapter in my

dissertation.

Finally, I would like to thank the other faculty and graduate students in the

philosophy department at UMass. I enjoyed every bit of the seminars and discussions in

this department. Especially, I want to express my gratitude towards Kirk Michaelian, who

became my closest friend.

vi

ABSTRACT


MAY 2010

NAMJOONG KIM, B.A., SOGANG UNIVERSITY

M.A., SOGANG UNIVERSITY

Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST

Directed by: Professor Phillip Bricker

About a decade ago, Adam Elga introduced philosophers to an intriguing puzzle.

In it, Sleeping Beauty, a perfectly rational agent, undergoes an experiment in which she

becomes ignorant of what time it is. This situation is puzzling for two reasons: First,

because there are two equally plausible views about how she will change her degree of

belief given her situation and, second, because the traditional rules for updating degrees

of belief don’t seem to apply to this case.

In this dissertation, my goals are to settle the debate concerning this puzzle and

to offer a new rule for updating some types of degrees of belief. Regarding the puzzle, I

will defend a view called “the Lesser view,” a view largely favorable to the Thirders’

position in the traditional debate on the puzzle. Regarding the general rule for updating, I

will present and defend a rule called “Shifted Jeffrey Conditionalization.”

My discussions of the above view and rule will complement each other: On the

one hand, I defend the Lesser view by making use of Shifted Jeffrey Conditionalization.

On the other hand, I test Shifted Jeffrey Conditionalization by applying it to various

credal transitions in the Sleeping Beauty problem and revise that rule in accordance with

vii

the results of the test application. In the end, I will present and defend an updating rule

called “General Shifted Jeffrey Conditionalization,” which I suspect is the general rule

for updating one’s degrees of belief in so-called tensed propositions.

viii

CONTENTS

Page

ACKNOWLEDGMENTS……………………………………………………….……….iv

ABSTRACT……………………………………………………………………………...vi

LIST OF TABLES………………………………………………………………………..xi

LIST OF FIGURES……………………………………………………………………...xii

CHAPTER

1. INTRODUCTION .................................................................................................. 1

A. Problem ....................................................................................................... 1

B. Goals ........................................................................................................... 3

C. Strategy ....................................................................................................... 5

D. Contents .................................................................................................... 13

2. UPDATING WITH A SINGLE OBSERVATION .............................................. 20

A. Introduction ............................................................................................... 20

B. Background ............................................................................................... 24

C. A Problem of the De Se Versions of SC and JC ....................................... 27

D. Shifted Jeffrey Conditionalization ............................................................ 30

E. Shifted Rigidity as a Conditional Expert Principle ................................... 37

F. Sleeping Beauty and Shifted Jeffrey Conditionalization .......................... 44

G. Conclusion ................................................................................................ 49

3. UPDATING WITH A SEQUENCE OF OBSERVATIONS ............................... 52

A. Introduction ............................................................................................... 52

B. Review of SC and SJC .............................................................................. 54

C. Strategy ..................................................................................................... 58

D. Updating with a Sequence of Observations .............................................. 60

E. A Defense of SSJC.................................................................................... 75

F. The SB Problem and the Inconsistency of SSJC ...................................... 81

G. A Diagnosis and a Potential Solution ....................................................... 82

H. Conclusion ................................................................................................ 91

4. UPDATING WITH DE PRIORI INFORMATION ............................................. 93

A. Introduction ............................................................................................... 93

B. Strategy ..................................................................................................... 94

C. Updating with De Priori Information ....................................................... 98

E. Temporal Conditional Multiple Expert Principle ................................... 111

F. A Defense of GSJC ................................................................................. 121

G. Too Far Past Does Not Matter ................................................................ 123

ix

H. Application to the SB Problem ............................................................... 127

I. The Relation between GSJC and Other Rules ........................................ 134

J. Conclusion .............................................................................................. 137

5. SATISFACTION OF DESIDERATA ................................................................ 139

A. Introduction ............................................................................................. 139

B. Background ............................................................................................. 140

C. Strategy ................................................................................................... 144

E. GSJC+ and GSR

+ ..................................................................................... 147

F. The Transitivity of GSR+ ........................................................................ 150

G. Synchronic and Diachronic Coherence ................................................... 157

H. Observational Exhaustiveness ................................................................ 162

I. Filling the Gap ........................................................................................ 165

J. Conclusion .............................................................................................. 168

6. CONCLUSION ................................................................................................... 169

A. Summary ................................................................................................. 169

B. Remaining Issue 1: Generalization for De Se Updating ......................... 171

C. Remaining Issue 2: Credence Distribution over the Partition ................ 176

D. Remaining Issue 3: The Possibility of a Rival Rule ............................... 181

E. Conclusion .............................................................................................. 187

APPENDICES

A. EQUIVALENCE BETWEEN GSJC- AND GSJC. ................................................... 189

B. EQUIVALENCE BETWEEN GSR+ AND GSJC

+ .................................................... 194

C. TRANSLATION BETWEEN TEMPORAL CONTEXTS ....................................... 198

D. EQUIVALENCE BETWEEN GSJC+ AND GSJC

0 .................................................. 206

BIBLIOGRAPHY ........................................................................................................... 213

x

LIST OF TABLES

Table Page

1: The Change of SB’s Credence in H………………………..…………………………...3

xi

LIST OF FIGURES

Figure Page

1: Update in Accordance with SJC…………………………………...………………….15

2: Update in Accordance with SSJC……………………………………………………..16

3: Update in Accordance with GSJC………………………………………………...…..18

4: Transitivity of GSJC………………………...………………………………………...19

5: Rain and Precipitation…………………………………………………………………42

6: Evidential Uncertainty in Sequential Updating…………...…………………………..71

7: Temporal Uncertainty in Sequential Updating…………………………...…………...73

8: Flying Birds, Running Animals, and Earthquake……………………………………..79

9: Deference by SBR and Temporal Ignorance………...…………………………...…...89

10: Reindexicalization and Deindexicalization 1………………………………………107

11: Reindexicalization and Deindexicalization 2…………………...………………….109

12: Judgmental Dependence………………………………………………...………….111

13: Direct and Indirect Data Providers…………………………………...…………….113

14: Referents of Indexicals in Different Temporal Contexts…………………………...199

15: Indexicals, Observations, and Intervals 1………………………………...………...200

16: Indexicals, Observations, and Intervals 2………………………...………………...201

17: Indexicals, Observations, and Intervals 3………………………...………………...202

1111

CHAPTER I

INTRODUCTION

A. Problem

In his “Self-locating Belief and Sleeping Beauty,” Elga (2000) presents an intriguing

puzzle: On Sunday night, Sleeping Beauty (hereafter: SB) knows that she will go through

the following experiment. On that night, she is put to sleep by a group of evil

experimenters. Then, they toss a fair coin. Case 1: (H) The coin lands heads. In this case,

the experimenters wake her up only once, on Monday. Case 2: (T) The coin lands tails. In

this case, they wake her up twice, the first time on Monday and the second time on

Tuesday. Between the two awakenings, they inject SB with a drug that erases her

memory of the first awakening. In either case, one minute after she wakes up on Monday,

she is told that it is Monday, and, when the experiment ends on Wednesday, she is

awakened with her memory of the last awakening intact. The puzzle ends with two

questions: When SB wakes up on Monday, what is her degree of belief in H? When she is

told that it is Monday, what is her degree of belief in H?1

There have been two dominant answers to the first question in the literature.

Halfers argue that the answer is 1/2 (Lewis 2001; Bradley 2003; Jenkins 2005): On

Monday, she wakes up with Sunday as her last memory. Since she fully expected to wake

up in that way, SB receives no new evidence relevant to H at that moment. Intuitively, a

1 This version of the SB problem is closer to Lewis’s (2001) version because the step of telling SB the day

is omitted in Elga’s original version (2000). In Chapter II, I will discuss Elga’s (simpler) version, to focus

on the first question, and in Chapter III and later, I will use Lewis’s (more complex) version, to answer the

second question.

2222

rational agent changes her degree of belief (hereafter: credence) in a proposition X only

when she receives new evidence relevant to X. SB’s credence in H was 1/2 on Sunday

night. Therefore, her credence in H on Monday is also 1/2. (Lewis 2001, p. 174.)

Thirders, on the other hand, contend that the answer is 1/3 (Elga 2000; Dorr 2002;

Weintraub 2004). According to their view, when SB wakes up on Monday, she knows

that she is waking up on either Monday or Tuesday but does not know which day it is. On

the one hand, she assigns 1/2 to H conditional on the possibility that she is waking up on

Monday. For remember that on Sunday night, she assigned 1/2 to H conditional on her

waking up on Monday. On the other hand, she assigns 0 to H conditional on the

possibility that she is waking up on Tuesday. This is because SB knows that if she is

waking up on Tuesday, the coin has already landed on tails. It follows that her actual

credence on Monday in H will be the weighted average between 1/2 and 0, where the

weights come from her credences that she is waking up on Monday and that she is

waking up on Tuesday. Since she cannot be sure that she is waking up on Monday, her

rational credence in H is less than 1/2. If we take symmetry into consideration, it will be

1/3. (However, many philosophers doubt that symmetry can restrict an agent’s credence

in this way. For the purposes of my argument, the exact value of SB’s credence on

Monday in H is not important. The important element of the Thirders’ view is that SB’s

credence in H is less than 1/2 when she wakes up on Monday.) (Elga 2000, pp. 144-145.)

On the second question, most Halfers and Thirders agree that when she learns

that it is Monday, SB's credence in H increases, but they disagree about the precise value

of the resulting credence in H. Many Halfers believe that it is 2/3, and virtually all

Thirders believe that it is 1/2.

3333

To compare their views, look at Table 1:

Table 1: The Change of SB’s Credence in H

Sunday night Monday morning told “Monday”

Halfers 1/2 1/2 2/3

Thirders 1/2 1/3 1/2

Ever since the publication of Elga’s paper, philosophers have debated between these two

options. So which side is right?

B. Goals

In this dissertation, I pursue two goals: First, I shall offer a solution to the SB problem;

basically, I shall defend the Thirders’ view minus any use of symmetry. Second, I shall

develop a general rule for updating de nunc credences; in other words, I shall discuss a

method for updating degrees of belief in tensed propositions.2

As the SB problem has attracted so many philosophers’ attention, my pursuit of

the first goal hardly requires any explanation. But why do I pursue the second? I do so

because it offers the straightest solution to the SB problem.

To appreciate this point, think about these facts: We know SB’s credence

distribution on Sunday night fairly well. For instance, we know that her credence in H on

that night is 1/2, we know that she knows then that it is Sunday, etc. Thus, finding her

2 By “de nunc credence,” I mean one’s degree of belief in a tensed proposition. By “tensed proposition,” I

mean a proposition-like entity that may have different truth-values relative to time, but not relative to the

individual. For instance, “it is raining now in Boston” may have different truth-values on Monday and

Tuesday, but if it is true/false on Monday for someone, then it is true/false on Monday for everyone.

4444

credence in H on Monday must be simply a matter of applying an appropriate updating

rule to SB’s credal transition from Sunday night to Monday morning. Therefore, if we

have a correct rule for updating applicable to this transition, it will be easy, in principle,

to calculate her credence in H on Monday.

Nevertheless, this approach has not been taken by many philosophers. Why? The

traditional updating rule, called “Strict Conditionalization,” states that an agent’s

credence at t in a proposition X is her conditional credence at t′<t in X given E, where E is

the totality of her observations during (t′,t]. Unfortunately, this rule does not apply to

SB's credal transition from Sunday night to Monday morning. SB learns (W) “SB wakes

up today with the memory of Sunday as the last memory remembered” on Monday.

However, on Sunday SB must have known (W′) “SB woke up today with the memory of

Saturday as the last memory remembered.” Since W is logically incompatible with W′,

her conditional credence on Sunday night in H given W has no defined value. Therefore,

Strict Conditionalization fails to provide a defined value for the former credence.

Hence, the most effective way to solve the SB problem is to apply an appropriate

updating rule to SB’s transition from Sunday night to Monday morning, but the Strict

Conditionalization rule is inappropriate here. While many other philosophers have tried

to solve the problem by appealing to other considerations for this difficulty (e.g. Lewis

appeals to his Principal Principle to settle the debate (Lewis 2001), Kierland and Monton

resort to the principle of minimizing inaccuracy of one's credence (Kierland and Monton

2005), etc.), I believe that finding the correct rule for de nunc updating is the best way to

settle the debate.

5555

In summary, I pursue two goals: Solving the SB problem and finding the general

rule for de nunc updating. I have suggested that achieving the second goal will be the

quickest way to achieve the first, but this does not mean that the second goal only has a

derivative value. Indeed, it is the opposite: The SB problem is so interesting precisely

because it reveals the fact that a new rule for updating is necessary to deal with de nunc

credences properly.

C. Strategy

As I said above, I intend to develop a new rule for updating de nunc credences. To this

end, I have developed the following criteria for an acceptable rule:

Solution I want my updating rule to provide intuitive answers to the two

questions of the SB problem.

Versatility I want my updating rule to apply to as many types of updating

situations as conceivable.

Coherence I want my updating rule to provide coherent results.3

How can I develop a de nunc updating rule that satisfies these criteria? I start by

reviewing the distinction between updating and revision, an established distinction in the

literature of qualitative belief change.4

3 Here, I am not using "coherent" in the technical sense that a rational agent's credence should not lead her

to a Dutch book or money pump. Rather, I am using that word to mean freedom from incoherence, where

incoherence is defined to be logical or conceptual inconsistencies. Bricker (ms.) says: “One way for a

theory to be internally coherent is for it to be logically inconsistent, but I suppose there are other ways.

Moreover, if a theory is unfaithful to the notions it aims to elucidate, be they notions of ordinary or

scientific thought, that too is a form of incoherence.” (Bricker ms., p. 1.)

6666

Revision An agent revises her beliefs because she has acquired better information

about what the world is like.

Updating An agent updates her beliefs because she has noticed that the world has

changed.

For instance, compare the following ways in which Jane comes to know that Barack

Obama won the presidential election in 2008: First, it is 2009 and Obama has been

president for some time, but Jane has come to know this fact today. Second, it is the

morning after the day of the 2008 election, and Jane has come to know that he became

the winner of the election last night. The former example is a case of revision, while the

latter is a case of updating.

To see why this distinction is important, first look at this formulation of Strict

Conditionalization: for any proposition X,

(1) Ct(X)=Ct′(X/E)

where Ct and Ct′ are the agent’s credence functions at t and t′<t, and E is the totality of

her observation made during (t′,t]. Now, note that (1) entails:

(2) Ct(X/E)=Ct′(X/E)

4 For a general theory of qualitative belief change (called “the AGM model"), see Gardenfors et. al. (1985).

For the distinction made here, see Katsuno and Mendelzon (1992).

7777

where X and E are genuine propositions (or proposition-like entities whose truth-values

are fixed). This means that if an agent obeys Strict Conditionalization, she comes to

preserve her past conditional credences. As Christensen points out (2000), this is a form

of epistemic conservatism:

The reasonableness of attractive instances of conditionalization seems to flow directly from the

reasonableness of maintaining the relevant conditional degrees of belief. And these conditional

degrees of belief are valuable because they reflect past learning experiences. (Christensen 2000, p.

354)

In this sense, epistemic conservatism is a good thing because one has worked hard to

acquire valuable information about the world and to incorporate such information into

one’s belief state in the form of conditional credence.

But what if the world changes? Note that from an agent’s present point of view,

her past credence in X given Ei is the measurement of how probable it was that X was true

then given that Ei was true then. In this sense, her past conditional credences are outdated.

Consequently, epistemic conservatism is a disaster in this case: although the agent’s past

conditional credence in X given E was her degree of belief in X's then truth given E's then

truth, she came to preserve it as her degree of belief in X's truth now given E's truth now.

This is clearly unreasonable unless the agent has a reason to believe that the world is

likely to stay in the same state as before. I see no reason to have such a belief about the

world.

8888

Note that this was not a problem in the traditional version of Strict

Conditionalization, because that rule concerns only de dicto credences and evidence—the

degrees of belief in propositions with fixed truth-values and in evidence whose content’s

truth-value is similarly fixed. Still, the following variant of Strict Conditionalization

appears to be incorrect: for any tensed proposition X,

(3) Ct(X)=Ct′(X/E)

where Ct′ and Ct are B's credence functions at t and t′<t, and E is the totality of her

observations during (t′,t]. (3) implies

(4) Ct(X/E)=Ct′(X/E)

where X and E are tensed propositions. (4) is epistemologically dangerous for the reason

explained above, which suggests that although Strict Conditionalization is a proper rule

for the revision of de dicto credences, it is not a proper rule for the updating of de nunc

credences.

Think about this: In the past, you had conditional credences about what would

happen in a future time v given that you would observe E in v, and, from your present

point of view, it might be v now.5 Given this fact, if you know your present temporal

location, then the following principle seems to capture a correct form of epistemic

5 I am using “v” instead of “t” because I want to emphasize that the mentioned time is an inter“v”alized

time, rather than a momentary one.

9999

conservatism: Suppose that the agent B observes nothing during (t′,t), for some moments

t and t′<t. Then, for any proposition X,

(5) Ct(X)=Ct′(X is true in v/E is true in v)

where Ct′ and Ct are the agent’s credence functions at t and t′, E is the totality of the

agent’s observations made at t, and v is a temporal interval that B fully believes at t that

she is in.6

I believe that (5) is often a correct rule for updating de nunc credences. For if an

agent changes her de nunc credences obeying (5), then her resulting conditional credence

in X given E will be equal to her previous conditional credence in X’s truth in v given E’s

truth in v, where v is her present temporal location. For example, suppose that on

Tuesday Jake learns that there will be a form of precipitation today and sets his credence

in raining today to be equal to his conditional credence on Monday that it rains on

Tuesday given that there is a form of precipitation on Tuesday. In a sense, he preserves

his conditional judgment of how likely it is to rain on a day d given that there is a form of

precipitation on d, where d is the same day referred to on Monday as “Tuesday” and

referred to on Tuesday as “today.” In this case, it is intuitive that Jake has to update in

accordance with (5) because it is the best way to respect his past learning.

However, (5) is not general enough for our purpose, because it does not apply to

a case of temporal uncertainty such as the SB problem. We need another candidate for the

6 By “X is true in v,” I mean that X is true at any moment in interval v. Likewise for “E is true in v.” Plus, I

assume that interval v is sufficiently narrow, but I do not try to provide a criterion for sufficient narrowness

here.

10101010

general rule for de nunc updating. Consider this one: Suppose that an agent B observes

nothing during (t′,t) for some momentary times t, t′ such that t′<t. Then, for any tensed

proposition X,

(6) Ct(X)=Σi∈JCt′(X is true in vj/E is true in vj)Ct(it is vj),

where Ct′ and Ct are B's credence functions at t and t′<t, E is the totality of the de nunc

content of the observation made at t, and vjj∈J is a partition of temporal intervals such

that B fully believes at t that she is in one of vjj∈J.

To me, (6) is a plausible generalization of (5). To see why, let vj1≤j≤n be a

partition of temporal intervals each of which B thinks at t to be possibly her then temporal

location. By (5),

Ct(X) would be

Cn(X is true in v1/E is true in v1) if B were sure at t that it is v1,

Cn(X is true in v2/E is true in v2) if B were sure at t that it is v2,

...

Cn(X is true in vn/E is true in vn) if B were sure at t that it is vn,

Since B does not know what time it is, it appears to be natural to take the weighted

average of the above values, with the weights coming from B's credence at t that it is vj.

Thus, (6).

Equipped with (6), we are ready to answer the first question of the SB problem:

Let s be SB's last conscious moment on Sunday, m be the moment of wakeup on Monday,

11111111

and m+ be one minute after m when SB is told that it is Monday. Clearly, SB does not

make any observation during (s,m). By (6),

(7) Cm(H)=

Cs(H is true on Monday/W is true on Monday)Cm(it is Monday)+

Cs(H is true on Tuesday/W is true on Tuesday)Cm(it is Tuesday).

Since H is a genuine proposition and its truth-value is insensitive to time,

(8) Cm(H)=

Cs(H/W is true on Monday)Cm(it is Monday)+

Cs(H/W is true on Tuesday)Cm(it is Tuesday).

Since SB fully knew on Sunday that if she wakes up on Tuesday, then H is false,

(9) Cm(H)=Cs(H/W is true on Monday)Cm(it is Monday).

Since she fully expected on Sunday to wake up on Monday,

(10) Cm(H)=Cs(H)Cm(it is Monday).

Since her credence at s in H was 1/2 and she cannot be sure at m that it is Monday,

(11) Cm(H)=1/2Cm(it is Monday)<1/2.

12121212

This is a result favorable to the Thirder view and incompatible with the Halfer view.

Thus, (6) is not only a plausible principle for updating de nunc credences, but

also it provides the SB problem with a solution that is largely favorable to the Thirder

view. This means that it satisfies one of my criteria for an acceptable principle for de

nunc updating.7

However, there are two reasons to suspect that (6) is not the end of the story.

First, it follows from (6) that SB’s credence in H does not change from the moment of

wakeup on Monday to that of being told that it is Monday, but we have a reason to

consider this to be an incoherent result. To see the reason, first think about the following

instance of (5) (which is a special case of (6)): At m+, SB knows that it is Monday. Thus,

(12) Cm+(H)=Cm(H is true on Monday/MON is true on Monday),

where MON is the tensed proposition that it is Monday. Since H has a fixed truth-value

and she knows at m that MON is, of course, true on Monday,

(13) Cm+(H)=Cm(H)<1/2.

This result does not cohere with

7 Of course, this result may not be satisfactory to the Halfers. But I am focusing here on the facts that (i) the

provided solution is attractive to a large group of philosophers and (ii) it comes with an argument that

might be plausible even to some philosophers who initially opposed its conclusion.

13131313

(14) Cm(H/MON)=1/2,

which is provable from the very (6).8

To understand why, think about this matter from SB's point of view when she is

told that it is Monday. Previously, her credence in H was 1/2 given MON, and, by

learning that it is Monday now, she also learns that it was previously Monday. Intuitively,

her credence in H must increase back to 1/2 by this learning.

Second, (6) will not apply to a credal transition from t′ to t if the agent makes any

observation during (t′,t). For instance, SB experiences W when she wakes up on Monday;

thus, she makes a seemingly important observation between s (=the night on Sunday) and

m+ (=one minute after her wakeup on Monday). Although we can apply (6) to her credal

transition from s to m and to her credal transition from m to m+ (the second application is

seen in the problem discussed in the last paragraph), it would be better if we could

calculate SB's credence at m+ in H all at once from her credence distribution on Sunday

night. In this sense, (6) does not satisfy the criterion of versatility.

In sum, while I have a promising prototype for the general rule for de nunc

updating, it does not perfectly satisfy the three aforementioned criteria. My goal in this

dissertation is to find a general rule for de nunc updating which is fully versatile and

coherent, and which provides a fully intuitive solution for the SB problem.

D. Contents

This dissertation consists of six chapters:

8 Hint: Show first that Cm(H&MON)=Cm(T&MON). Since Cm(MON)=Cm(H&MON)+ Cm(T&MON),

Cm(H/MON)=Cm(H&MON)/Cm(MON)=Cm(H&MON)/2Cm(H&MON)=1/2. (See Chapter II for a full proof.)

14141414

Chapter I. Introduction

Chapter II. Updating with a Single Observation

Chapter III. Updating with a Sequence of Observations

Chapter IV. Updating with De Priori Information

Chapter V. Satisfaction of Desiderata

Chapter VI. Conclusion

The titles of Chapters I and VI are self-explanatory. Roughly, the main body of the

dissertation consists of three parts, each devoted to one of my criteria: (i) In Chapter II, I

discuss a relatively simple principle for de nunc updating, to solve the SB problem. (ii) In

Chapters III and IV, I generalize that simple principle into more versatile principles. (iii)

In Chapter V, I prove that the most general principle has several properties that we can

regard as forms of coherence. I provide more details below.

In Chapter II, I will discuss how a rational agent changes her credence in a

tensed proposition from t to t′, assuming that she receives no evidence during (t′,t). I will

present and defend the following principle for updating de nunc credences, which I call

“Shifted Jeffrey Conditionalization" or “SJC": Let X be any tensed proposition. Then,

roughly,

(15) Ct(X)=Σi∈I,j∈JCt′(X is true in vj/Ej is true in vj)Ct(Ej is true&it is vj),

15151515

where Ct′ and Ct are B's credence functions at t and t′, Eii∈I is a partition whose member

represents an observation she might be making at t, and vjj∈J is a partition whose

member represents a temporal interval that she might be in at t.9

Since (6) is a special case of SJC (or (15)), SJC suffers from the two problems I

discussed earlier. Figure 1 is a diagram showing how SB's credence in H changes in

accordance with SJC:

Figure 1: Update in Accordance with SJC. Not as versatile as wanted. Some incoherent

result.

where x is some value less than 1/2. First, SJC does not apply to the all-at-once updating

from s to m, which makes it not as versatile as we want. Second, SJC yields the counter-

intuitive result that her credence in H does not increase back to 1/2.

In Chapter III, I discuss how an agent can change her credence in a tensed

proposition from t′ to t, assuming that she makes a finite sequence of observations during

(t,t]. I will present a rule governing this type of updating, which I call “Sequential Shifted

Jeffrey Conditionalization” or “SSJC.” I cannot provide a proper formulation of this rule

here; simply, we do not have the necessary formal and conceptual resources yet. Instead,

9 Note that (15) is more general than (6) in that (15) incorporates uncertainty about what observation was

made as well as uncertainty about what time it is.

16161616

I provide its instance involving how SB's credence in H changes from the night on

Sunday (=s) to when she is told that it is Monday (=m+):

(16)

×

=

+

+

Monday isit and trueis presently, (d)

&Monday it was and true was ,previously (c)

(

Mondayon trueis (b)

&Mondayon trueis (a)

Monday/ on trueis (

)(

MON

W

C

MON

W

HC

HC

ms

m.

SB makes an observation twice during (s,m], the first time W and the second time MON.

In other words, she makes a sequence of observations <W,MON> during (s,m]. Here is

the core idea of (16): To find SB’s rational credence in H given this sequence of

observations, we need to specify, for each element of this sequence, the time of her

observing it, as you see in (c) and (d), and figure out her prior credence in H given that

each element of the sequence is true at those specified times, as you see in (a) and (b).

Hopefully, the reader will see how to generalize this idea into a formal rule for updating.

Unfortunately, SSJC is inconsistent. Figure 2 is a diagram showing how SB's credence in

H changes in accordance with SSJC:

Figure 2: Update in Accordance with SSJC. Versatile but inconsistent because x≠1/2.

where x is some value less than 1/2. As you see above, SSJC provides a different result

depending upon whether we apply it to the transition from s to m+ or to the transition

17171717

from s to m and then to that from m to m+. This makes it an unacceptable rule (unless

restricted by a suitable proviso).

In Chapter IV, I discuss how a rational agent changes her credence in a tensed

proposition from t′ to t, assuming that her observation may include information about her

temporal location at t′ or an earlier moment. The updating rule presented in this chapter

will be the most general updating rule discussed in this dissertation. I will call this rule

“General Shifted Jeffrey Conditionalization” or “GSJC.” Again, I do not try to present

the rule here. Instead, I discuss how it solves the problem that SB's credence in H does

not change at m+ although her then evidence MON is intuitively relevant to H. Under

several assumptions, we can derive the following claim from GSJC:

(17)

×

=

+

+

Monday it was ,previously (d)

&Monday isit and trueis presently, (c)

(

Monday isit presently, (b)

&Monday on trueis (a)

Monday/ on trueis (

)( MON

C

MON

HC

HC

mm

m.

In this updating process, MON is the only thing that SB has learned during (m,m+]. In a

trivial sense, we can say that <MON> is the sequence of observation that she has made

during (m,m+].

Here is the core idea in (17): To find her rational credence in H given this

sequence of an observation, we need to specify the time of her observing W and figure

out her prior credence in H given W’s truth at the specified time (which happens to be

Monday), as you see in (a) and (c). But that’s not all. We also need to specify what time it

was before she observed W and figure out her prior credence in H given W’s truth on

18181818

Monday plus the prior time’s being Monday, as you see in (b) and (d). I will call such

information about what time it was at the prior time “de priori information.”10

Let us see how this modification provides a better model for how SB's credence

in H changes. According to GSJC, SB’s credence in H changes as in Figure 3:

Figure 3: Update in accordance with GSJC. Versatile, coherent, providing results

compliant with the popular Thirder view.

where x is some value less than 1/2. As Figure 3 demonstrates, GSJC provides a coherent

and intuitive model for the change of SB's credence in H. This model also complies with

the view of Thirders, regarding both questions asked earlier.

In Chapter V, I argue that GSJC has several properties desirable for any cogent

rule for updating. In particular, I will show that (i) GSJC can be regarded as a binary

relation between the given agent’s credence functions (at different times) and (ii) as such,

GSJC is transitive, if all of the agent’s credence functions are synchronically coherent.

See Figure 4:

10 I owe this term to Gareth Matthews.

19191919

Figure 4: Transitivity of GSJC. Here, I(t,t′) is information observed during (t,t′].

Put another way, GSJC provides the same result whether you update all at once from t1 to

tn or step-by-step from t1 to t2, t2 to t3, … to tn.

In Chapter VI, I discuss (i) how to modify GSJC into a general rule for de se

updating, (ii) how to overcome a potential problem of GSJC, and (iii) whether there is

any comparably plausible but simpler rule for updating. I conclude that GSJC is likely to

be identical or very close to the general rule for de se updating.

Through these discussions, I will defend my view that GSJC is the rational rule

for updating de nunc credences. I will argue that GSJC provides not only an ideal

solution for the Sleeping Beauty problem but also a versatile and coherent general rule

for de nunc updating.

20202020

CHAPTER II

UPDATING WITH A SINGLE OBSERVATION11

A. Introduction

SB problem 0. Suppose that Sleeping Beauty (hereafter: SB), a paragon of probabilistic

rationality, knows the following facts on Sunday: A group of evil experimenters will put

her to sleep on that day. Next, they will toss a fair coin. Case 1: (H) The coin lands heads.

In this case, the experimenters will wake SB only on Monday. Case 2: (T) The coin lands

tails. In this case, they will wake SB for the first time on Monday, inject her with a drug

that erases her memory of Monday, and then wake her for the second time on Tuesday. In

either case, the experiment is over on Wednesday.

For brevity, let s be the last moment on Sunday at which SB is conscious and let

m be the moment of waking up on Monday. Accordingly, let Cm and Cs be her credence

functions at m and s. The question is: “What is SB’s credence at m in H?” There have

been two dominant answers: According to the Thirder view, Cm(H)=1/3 (Elga 2000).

According to the Halfer view, Cm(H)=1/2 (Lewis 2001). Both views have good

arguments in their favor.

Halfers contend: Let W be that SB wakes up today with the memory of Sunday

as the last memory. When SB was put to sleep on Sunday, she fully expected to receive

W as her next evidence. Furthermore, that she has awakened today with such and such

11 This chapter is identical to my paper published in Synthese (Kim 2009) except for several changes of

notation and correction of typos.

21212121

memory seems irrelevant to whether the coin lands heads or tails. Hence, W is neither

new nor relevant to H. But the thesis below derives from the standard rule for credence

updating:

(1) If no new evidence relevant to X is received,

then it is irrational for an agent to change her credence in X.

Thus, SB doesn’t change her credence in H after waking up on Monday. It is not

controversial that SB assigns the credence of 1/2 to H on Sunday night. Therefore, she

assigns 1/2 to H after waking up on Monday. (Lewis 2001, 174.)

Thirders argue: Let MON be that it is Monday, and TUE be that it is Tuesday.

Then, we can define H1, T1, and T2 as below:

H1: H&MON

T1: T&MON

T2: T&TUE

Obviously, these exhaust the possibilities open to SB when she wakes up on Monday. On

the one hand, suppose that SB was immediately told that it’s Monday after waking up on

Monday. In this scenario, she would assign the same credence of 1/2 to H and T. Hence,

Cm(H/MON)=1/2=Cm(T/MON). It follows that

(2) Cm(H1)=Cm(T1).

22222222

On the other hand, assume that SB was told immediately after waking up on Monday that

the coin landed on tails. The evidence she receives on Monday, in this scenario, is

compatible with either MON or TUE; for, if the coin lands tails, she wakes up both on

Monday and Tuesday. By a principle of indifference, it seems rational to assign the same

credence to MON and TUE, given T; formally, Cm(MON/T)=Cm(TUE/T). It follows that

(3) Cm(T1)=Cm(T2).

In sum, Cm(H1)=Cm(T1)=Cm(T2)=1/3. But H1 is the only possibility in which the coin

lands heads. Hence, Cm(H)=1/3. (Elga 2000, 143-144.)

Which side made a mistake? Theses (1) and (2) contradict each other.12

Hence,

Halfers, who accept (1), are bound to reject (2), and Thirders, who accept (2), are

committed to the rejection of (1). Thus, each side complains that the other’s argument is

factually incorrect. The problem is that neither side has been able to explain why the

other side’s key premise is wrong. For this reason, the debate has continued. I suspect,

however, that both sides have missed an important point. We know quite well SB’s belief

state on Sunday night; on that night, her credence in H was 1/2, and she knew how the

experiment would proceed. W is the only evidence she acquires until she wakes up next

morning. Isn’t it then simply a matter of applying Strict Conditionalization (hereafter:

SC), the traditional principle for updating credences, to SB’s credal transition from s to m?

12 Suppose theses (1) and (2). Hence, Cm(H)=Cs(H)=1/2 and Cm(H1)=Cm(T1). Unless Cm(T2)=0,

Cm(H1)<Cm(T1)+Cm(T2). Since H1 exhausts the H-possibility and T1 and T2 exhaust the T-possibilities,

Cm(H)<Cm(T). Therefore, 1/2=Cm(H)<1/2. Done.

23232323

Unfortunately, the problem is not that simple. It is an instance of SC that

Cm(H)=Cs(H/W). However, SB fully knew on Sunday night that she did not wake up on

that day with the memory of Sunday as the last memory, and so Cs(H/W) is undefined.

Nevertheless, I believe that there exists an updating principle applicable to SB’s

credal transition from s to m. My first goal in this chapter is to find an updating rule for

when the domains of an agent’s credence functions include tensed propositions or

proposition-like entities whose truth-values are possibly different depending upon the

time of evaluation. I will call this new updating principle “Shifted Jeffrey

Conditionalization” (hereafter: SJC). I will argue that because W is a tensed proposition,

SB has to use SJC, not SC, for her credal transition from s to m.

My second goal in this chapter is to explore the ramifications of this new

updating rule concerning the SB problem. I shall make three claims: First, thesis (1) is

disprovable under SJC. Second, thesis (2) is provable under SJC. Third, thesis (3) is

neither provable nor disprovable in any obvious way.

If these claims are true, who wins in the Sleeping Beauty debate? On the one

hand, Halfers are clearly not the winners. For their argument is unsound if thesis (1) is

false and SB’s credence at m in H is less than 1/2 if thesis (2) is true. On the other hand,

this does not necessarily mean a victory for the Thirders. For thesis (3) is an essential

element of their view but SJC does not obviously support it.

Consequently, I partially accept the Thirder view but take a more lenient position:

I will argue that SB’s credence in H is less than 1/2 when she wakes up, but I will remain

silent about what the value should be. Call this “the Lesser view.” In this chapter, I will

defend it.

24242424

I will proceed in the following order: In Section B, I will clarify my assumptions

and terminology and review traditional updating principles. In Section C, I will argue that

those updating principles do not work for beliefs and evidence whose truth-values are

different relative to time and/or individual. In Section D, I will present an alternative

updating principle, SJC, for such beliefs and evidence. In Section E, I will defend SJC by

extending Gaifman’s influential view of expert principles. Finally, in Section F, I will

apply SJC to the SB problem. As a result, the Halfers’ thesis (1) will be criticized and the

Thirders’ thesis (2) will be defended.

B. Background

In this section, I will clarify my assumptions and terminology about beliefs, contents, and

credences.

First, belief is a relation between an agent and a proposition-like entity. For

example, consider Jane’s belief that (C) Caesar crossed the Rubicon in 49 BC. According

to my assumption, this belief is a relation between Jane and C.

Second, the truth-values of some beliefs remain the same whoever has them or

whenever they are had. Jane’s belief in the above paragraph is a good example. Let’s call

such a belief a “de dicto belief” and its content a “(genuine) proposition.” I consider such

a belief to be purely about which possible world the agent is located in.

Third, the truth-values of some beliefs are different relative to times and/or

individuals. For example, consider Jane’s belief expressed by “I am 15 years old.” The

content of this belief will be true of anyone who is 15 years old, but won’t be true of

anybody who is younger or older. Let’s call such a belief an “irreducibly de se belief”

25252525

and its content an “irreducibly centered proposition.” I consider a belief of this type to be

at least partially about what time it is and/or who the agent is.

Fourth, I will call any de dicto or irreducibly de se belief simply “a de se belief”

and its content “a centered proposition.”13

Fifth, belief is not all-or-nothing but comes in degrees. Degrees of belief, called

“credences,” are probabilities in that they satisfy Kolmogorov’s three axioms: Non-

Negativity, Normality, and Additivity.

Sixth, I will call the degree of a de dicto belief a “de dicto credence,” that of an

irreducibly de se belief an “irreducibly de se credence,” and that of a de se belief a “de se

credence.”

So far, I have clarified my assumptions and terminology. Now, to the question:

“What is the correct rule for updating de se credences?” To answer, it is a good idea to

review the traditional rules for updating de dicto credences.

First, how is a rational agent supposed to update her de dicto credences given

certain evidence? Consider the strongest proposition E such that a rational agent B

becomes certain at tn+1 of E, as a result of her experience at tn+1.14

Then, B should update

by Strict Conditionalization: (SC) for any proposition X, Cn+1(X)=Cn(X/E)=df

Cn(X&E)/Cn(E), where Cn(E)≥0. To see how SC works, consider this example: Example

13 I am following Lewis (1979) in defining de se belief in this way: “I say that all belief is ‘self-locating

belief.’ Belief de dicto is self-locating belief with respect to a logical space; belief irreducibly de se is self-

locating belief at least partly with respect to ordinary time and space, or with respect to the population”

(Lewis 1979, 522).

14

For simplicity, I will assume that the given agent makes observations at only a countable number of

moments in her life. Let’s call them “epistemic moments.” From now on, I will use “tα” to refer to a series

of epistemic moments, where α indicates the order and contiguity of those moments. Hence, for any

m,n∈N, tm is later than tn iff m>n, and for any n∈N, tn+1 is the epistemic moment next to tn. In addition, I

will use “Cα” to refer to the given agent’s credence function at tα.

26262626

1. Jane’s previous conditional credence in a coin’s landing heads was 3/4 given that it is

tossed. After she receives the evidence that it was tossed, her present unconditional

credence in the coin’s landing heads becomes 3/4.

Second, how does a rational agent update her de dicto credences if no

proposition meets the condition satisfied by E in the last paragraph? Suppose that Eii∈I

is a partition such that for any i∈I, an agent B’s credence in Ei is directly set by her

experience at tn+1. I will call each Ei “(B’s) observation proposition at tn+1.” Let Eii∈J be

a subset of this partition (so J⊆I) such that Cn+1(Ei)>0 for any i∈J. If also Cn(Ei)>0 for

any i∈J, then we call Eii∈J “(B’s) observation partition at tn+1.” In such a case, Richard

Jeffrey suggests that B should update her de dicto credences by Jeffrey Conditionalization:

(JC) for any proposition X, ∑∈

++ =Ji

ininn ECEXCXC )()/()( 11 (Jeffrey 1990, pp. 164-83).

To see how JC works, consider this example: Example 2. At 2:00 PM, Jane is looking at

a piece of vegetable under a dim light, uncertain whether it is green or violet. Hence, she

is uncertain about which of G (“this piece of vegetable is green”) and V (“this piece of

vegetable is violet”) is true. Still, her experience somehow influences her credences in G

and V; consequently, her credences at 2:00 PM in G and V are 0.3 and 0.7. Then, what

should her credence be in C (“it is a piece of cabbage”)? Her conditional credence at 1:59

PM in C was 0.6 given G and 0.2 given V. By JC, C2:00 PM(C)=C1:59 PM(C/G)C2:00 PM(G)+

C1:59 PM(C/V)C2:00 PM(V)=0.32.

If, as I have demonstrated, SC or JC is the rule for updating de dicto credences,

what then is the rule for updating de se credences? According to David Lewis, we can

easily find a candidate for such a rule: just replace “de dicto” with “de se” and

27272727

“proposition” with “centered proposition” in SC (Lewis 1979, 534). Another candidate

can be found by carrying out the same replacement in JC.

However, I believe that the de se versions of SC and JC are incorrect. For they

have a common problem, which I discuss in the next section.

C. A Problem of the De Se Versions of SC and JC

I want to show that the de se versions of both SC and JC are untenable. However,

criticizing them will be a tedious job if done one by one. A more efficient method will be

to first find a thesis common to the two principles and, second, show that this common

thesis has a fatal problem. Is there such a thesis?

According to Richard Jeffrey (1984, p. 135), we can easily prove this:

(C) Suppose that both Cn and Cn+1 satisfy Kolmogorov’s axioms. Let E be the

agent’s total evidence at tn+1. Then, (a) for any proposition X, Cn+1(X)=Cn(X/E) iff

(b) Cn+1(E)=1 and (c) for any proposition X, Cn+1(X/E)=Cn(X/E).

In other words, an agent is a strict conditionalizer iff she certainly believes her total

evidence, and its probabilistic relevance to any other belief is unchanged by updating.

Following Jeffrey’s terminology, let’s call condition (c) “Rigidity.” We can also prove

this broader claim (Jeffrey 1984, p. 136):

(K) Suppose that both Cn and Cn+1 satisfy Kolmogorov’s axioms. Let Eii∈I

be the agent’s observation partition at tn+1. Then, (d) for any proposition X,

28282828

∑∈

++ =Ii

ininn ECEXCXC )()/()( 11 iff (e) for any proposition X and i∈I,

Cn+1(X/Ei)=Cn(X/Ei).

In other words, an agent is a Jeffrey conditionalizer iff her old and new credence

functions satisfy Rigidity. According to (C) and (K), the truth of Rigidity regarding each

member of her observation partition is a common necessary condition of SC and JC.15

Now, replace “proposition” with “centered proposition” in (C) and (K); still, they are

provable claims. Let’s call the results of this replacement within (a), (d), and (c)/(e) “the

de se versions of SC, JC, and Rigidity.” Then, the de se version of Rigidity is the

common necessary condition of those of SC and JC.

But Rigidity has a fatal flaw. It conflicts with the logic of de se beliefs,16

in that a

probabilistic updating pattern it supports often leads to deductively invalid reasoning.

Think about this example: Example 3. Let R be the centered proposition expressed by “it

is raining now,” and P be the one expressed by “some form of precipitation is occurring

now.” Assume that P, not-P is Jake’s observation partition at 2:00 PM. For our purpose,

it is best to discuss the present example from the first-person point of view; hence, we let

Cprev be Jake’s credence function at 1:59 PM and Cnow be his credence function at 2:00

15 If the agent has certain total evidence E, then E is the sole member of her observation partition.

16

Meacham (2008) provides a simpler argument against the de se version of SC: Once a de se SC-er

becomes certain that it is 9:00 AM, she cannot abandon that belief at 9:01 AM. Since such abandonment

occurs too often, the de se version of SC cannot be true. This argument seems to be sound, but I have two

reasons to look for an alternative. First, Meacham’s argument does not show the incorrectness of the de se

version of JC. Second, his argument cannot defeat the de se version of SC formulated in terms of primitive

conditional credences. For the problem raised by Meacham is a form of the zero-denominator problem,

which can be avoided if we define unconditional credence in terms of conditional credence rather than

doing the opposite (Hajek 2003).

29292929

PM. Additionally, assume that Cprev(P)=0.5 and Cnow(P)>0. Obviously, we can derive this

fact from the de se version of Rigidity:

(4) If Cprev(R/P)≈1, Cnow(R/P)≈1.17

From the suppositions, we can easily show that (i) if Cprev(P⊃R)≈1, then Cprev(R/P)≈1 and

(ii) if Cnow(R/P)≈1, then Cnow(P⊃R)≈1.18

Hence, this is true:

(5) If Cprev(P⊃R)≈1, then Cnow(P⊃R) ≈1.

This means that the following normative sentence is true of Jake at 2:00 PM:

(6) If I strongly believed P⊃R previously,

I must strongly believe P⊃R now.

Practically, following (6) amounts to doing this type of reasoning:

(7) It was previously the case that, if P, then R.

Therefore, it is now the case that, if P, then R.

17 Here, “≈” is a synonym of “is almost identical to.”

18

Here, “⊃” is the material implication connective. To prove (i), suppose that Cprev(P⊃R)≈1. We know that

Cprev(P⊃R)=Cprev(not-(P&not-R))=1-(1-Cprev(R/P))Cprev(P). By the assumption that Cprev(P)=0.5,

Cprev(P⊃R)=1/2(1+Cprev(R/P)). Thus, Cprev(R/P)=2Cprev(P⊃R)-1. By supposition, Cprev(R/P) ≈1. To prove

(ii), suppose that Cnow(R/P) ≈1. Since Cnow(P⊃R)=1-(1-Cnow(R/P))Cnow(P), Cnow(P⊃R)≈1. Done.

30303030

However, this argument form is invalid. For it is surely possible that the premise is true

but the conclusion is false: It was previously raining, and it is now snowing. Hence, the

de se version of Rigidity may lead to invalid reasoning.19

This is a good reason to

abandon it.

Since the de se version of Rigidity is a common necessary condition of SC and

JC, we should reject both.

D. Shifted Jeffrey Conditionalization

If neither SC nor JC is an acceptable principle for de se updating, then what is? I do not

have a general principle for updating de se credences yet. In this section, however, I will

suggest a new principle for updating the degrees of some special de se beliefs.

Before presenting this new updating principle, I need new notions related to

beliefs, contents, credences, and time:

First, there are de se beliefs whose truth-values are different depending upon

when they are had but not upon who has them. For instance, think about Jane and Jack’s

common belief that it rains today in Boston. It can have different truth-values at different

times: It is possible that this belief is true on Monday but false on Tuesday. However, it

cannot have different truth-values for different people at any given time: It is impossible

that this belief is true for Jane but false for Jack on any given day. Let’s call a de se belief

of this type an “irreducibly de nunc belief” and the content of an irreducibly de nunc

19 Admittedly, this result is dependent upon the suppositions that (i) Cprev(P)=0.5 and (ii) Cnow(P)>0.

However, neither supposition includes anything that possibly justifies the reasoning of (7). For (i) merely

means that Jake is neutral between P and not-P and (ii) just means that he doesn’t rule out P.

31313131

belief an “irreducibly tensed proposition.” I consider a belief of this type to be partially

about what time it is.

Second, we will call a de dicto or irreducibly de nunc belief a “de nunc belief,”

and we will call a genuine or irreducibly tensed proposition a “tensed proposition.”20

Third, we will call the degree of an irreducibly de nunc belief an “irreducibly de

nunc credence” and that of a de nunc belief a “de nunc credence.”

Fourth, I introduce this definition: For any tensed proposition X and (temporal)

interval v,

The truth-value of X is invariant within v iff for any moments t and t′ in v, X’s

truth at t logically implies X’s truth at t′.

For instance, consider tensed proposition R expressed by “it rains in Boston at some time

today.” The truth-value of R is invariant within Monday; for, if R is true/false at some

moment on Monday, then R is true/false at any other moment on Monday. Similarly, the

truth-value of R is invariant within Tuesday. However, R can have different truth-values

on Monday and on Tuesday.

In addition to these notions, we need an adequate formal language to formulate a

new updating principle. Hence, I construct such a language L:

First, we need the traditional probability language’s logical connectives,

arithmetic operators, and probability function letters in our new language.

20 For the deductive theories of tensed propositions, see Prior (2003) or Rescher and Urquhart (1971).

32323232

Second, as we need propositional letters in the traditional probability theory, we

need tensed propositional letters in our new language. For this, we reserve “E,” “F,” “X,”

“Y,” and “Z.”

Third, we use “t” as a moment letter, which denotes a moment or a point-sized

temporal location, and “v” as an interval letter, which denotes an interval or a continuous

class of moments. Together, we call them time letters.

Fourth, given a time letter, we use the corresponding capital letter as a special

tensed proposition letter, denoting the tensed proposition that the present moment is or

belongs to the time denoted by the variable. For example, if “t” is the letter denoting the

moment of 9:00 AM on Sep. 4th

in 2006, then “T” is the letter denoting the tensed

proposition that the present time is exactly that moment. Similarly, if “v” is the interval

letter denoting the day of Sep. 4th

in 2006, then “V” is the letter denoting the tensed

proposition that it is that day.

Fifth, we introduce binary operators “at” and “in.” Let “ϕ,” “τ,” and “ν” be

schematic letters replaced with a tensed propositional letter, a moment letter, and an

interval letter, respectively. Here is a meaning schema for “at”:

“ϕ at τ” means that ϕ is true at τ.

For instance, let “R” mean the tensed proposition that it rains now in Boston and let “t”

denote the moment of 9:00 AM on July 18th

2006. Then, “R at t” denotes the proposition

that it rains in Boston at that moment. In addition, we introduce this abbreviation:

33333333

“ϕ in ν” abbreviates “(∀t∈ν)(ϕ at t).”

In other words, “ϕ in ν” means that ϕ is true throughout ν. For example, let v be July 18th

2006. Then, “R in v” means that it rains in Boston throughout July 18th

2006.

Now, we are ready to discuss a principle for updating de nunc credences. First,

we consider an agent who has a tensed proposition E as certain total evidence and fully

believes that she is located in v. For such an agent, I recommend this method of updating

her credence in a tensed proposition, which I call “Shifted Strict Conditionalization”:

(SSC) Cn+1(X)=Cn(X in v/E in v) if E is the agent’s certain total evidence at tn+1

and she fully believes at tn+1 that it is v,

where Ct(E in v)>0, and the truth-values of X and E are invariant within v. In other words,

if E is a rational agent’s certain total evidence at t+1, and she fully believes at tn+1 that it

is v, her credence at tn+1 in a tensed proposition X is the same as her previous conditional

credence in [X’s truth in v] given [E’s truth in v], as long as the conditional credence is

defined and neither X nor E can have different truth-values at any two moments in v.

In order to see how SSC works, consider this example: Example 4. Let P be that

some form of precipitation occurs today in Boston and R be that it rains today in Boston.

Suppose that on Sunday, Jane’s conditional credence in [R’s truth on Monday] given [P’s

truth on Monday] is 0.3.21

On Monday, she learns and becomes certain that some form of

precipitation occurs today in Boston, and today is Monday. What, then, is Jane’s rational

21 In this paper, I use brackets simply to avoid scope confusion.

34343434

credence on Monday in its raining today in Boston? My answer: CMON(R)=CSUN(R is true

on Monday/P is true on Monday)=0.3 by SSC.

Second, consider an agent B who possibly lacks certain total evidence or is

uncertain of what time it is. In order to capture such uncertainties, we assume two

partitions Eii∈I and vjj∈J, such that Eis are the tensed propositions, such that B’s

credences in Eis are directly set by her experience at tn+1, and such that vjs are temporal

intervals covering the minimal interval that B fully believes at tn+1 that she is located in.

Given these partitions, we will call Eis “(B’s) observation propositions at tn+1” and Vjs

“(B’s) temporal location propositions at tn+1.” Roughly, an agent lacks certain total

evidence iff she is uncertain of which of her observation propositions is true, and she is

uncertain of what time it is iff she is uncertain of which of her temporal location

propositions is true.

It’s possible to unify these two dimensions of uncertainty into one. Consider

Ei&Vj<i,j>∈I×J, consisting of consistent conjunctions of the members of the above two

partitions. Let’s call any such conjunction “(B’s) time-observation proposition at tn+1.” To

B, one of her time-observation propositions at tn+1 is a candidate for the true conjunction

of her observation and temporal location propositions at tn+1. Roughly, an agent lacks

certain total evidence and/or is not sure of what time it is iff she is uncertain of which of

her time-observation propositions is true.

Naturally, we focus upon B’s time-observation propositions whose truth she

doesn’t completely rule out at tn+1. Thus, let K⊆I×J be the class of <i,j>s such that

35353535

Cn+1(Ei&Vj)>0, where Cn+1 is B’s credence function at tn+1. If Cn(Ei at vj)>0 for any <i,j>

in K, I call Ei&Vj<i,j>∈K “(B’s) time-observation partition at tn+1.”

Finally, we formulate a method of updating credences in tensed propositions,

which I call “Shifted Jeffrey Conditionalization”:

(SJC) ∑>∈<

++ =Kji

jinjijnn VECvinEvinXCXC,

11 )&()/()( if E i& V j < i, j>∈ K

is the agent’s time-observation partition at tn+1,

where, for any i∈I and j∈J, the truth-values of X and Ei are invariant within vj. In other

words, if an agent lacks certain total evidence and/or is uncertain of what time it is at tn+1,

then her rational credence at tn+1 in a tensed proposition X is the weighted average of the

results of applying SSC to X with various time-observation propositions at tn+1, in which

the weights are her credences at tn+1 in the time-observation propositions, as long as

neither X nor any of Eis logically can have different truth-values at any two moments

within each interval vj.

To see how SJC works, think about the following examples: Example 5. Let “P”

and “R” have the same meanings as in Example 4. On Sunday, Jane’s conditional

credence in [R’s truth on Monday] given [P’s truth on Monday] is 0.5, and her

conditional credence in [R’s truth on Monday] given [not-P’s truth on Monday] is, of

course, 0. On Monday, some perfectly reliable person tells her that it is Monday and

something about today’s weather, but she doesn’t hear the latter information clearly. Has

he said that some form of precipitation occurs, or that it doesn’t occur, today in Boston?

She is unsure. Her credence in the former is 0.7 and that in the latter is 0.3. In this case,

36363636

what is Jane’s credence on Monday in its raining in Boston on that day? My answer:

CMON(R)=CSUN(R is true on Monday/P is true on Monday)CMON(P)+CSUN(R is true on

Monday/not-P is true on Monday)CMON(not-P)=0.35 by SJC.

Example 6. Again, keep the meanings of “P” and “R” the same. On Sunday,

Jane is put to sleep with one of two drugs. The first drug’s effect lasts for just one night,

making her wake up on Monday. The second drug’s effect lasts longer, making her wake

up on Tuesday. Her conditional credence on Sunday in [R’s truth on Monday] given [P’s

truth on Monday] is 0.8, and her conditional credence in [R’s truth on Tuesday] given

[P’s truth on Tuesday] is 0.2. On Monday, Jane is told by a perfectly reliable person that

some form of precipitation occurs today in Boston, and, not knowing which drug she took,

Jane assigns the credence of 0.4 to its being Monday and that of 0.6 to its being Tuesday.

In this case, what is Jane’s rational credence on Monday in R? My answer:

CMON(R)=CSUN(R is true on Monday/P is true on Monday)CMON(it is Monday)+CSUN(R is

true on Tuesday/P is true on Tuesday)CMON(it is Tuesday)=0.44 by SJC.

Despite the complicated appearance, SSC and SJC are actually quite intuitive.

Certainly given E as total evidence and v as her temporal location, an agent should assign

to X the previous conditional credence in X’s truth in v given E’s truth in v, not that in X

given E. For the latter conditional credence is the (previous) conditional credence in X’s

previous truth given E’s previous truth, which seems irrelevant to the (present) credence

in X’s present truth. Once accepting this, it seems natural that if the agent lacks certain

total evidence and/or does not know what time it is, the new credence in X should be the

weighted average of the previous conditional credence in [X’s truth in vj] given [Ei’s truth

37373737

in vj], where Eis and vjs are the candidates for the presently true observation proposition

and the temporal interval in which she is presently located.

In this section, I have presented a new principle for updating credences and

illustrated how that principle works with examples. The intuitive answers yielded by SJC

provide evidence in support of it, but additional argument is needed in order for us to

accept its validity.

E. Shifted Rigidity as a Conditional Expert Principle

In this section, first, I will present a principle entailing SJC (which has SSC as a special

case), and, second, I will discuss how that principle can be promoted by an intuitive

expansion of Gaifman’s Expert Principle (Gaifman 1988).

First, the principle: I call it “Shifted Rigidity.” Let Ei&Vj<i,j>∈K be the agent’s

time-observation partition at tn+1. Then, for any tensed proposition X and any <i, j> in K,

(SR) Cn+1(X/Ei&Vj)=Cn(X in vj/Ei in vj),

where the truth-values of Ei and X are invariant within each vj. From the agent’s point of

view, this says, “The present relevance of Ei to X, on the additional condition that I am

located in interval vj, is the same as the previous relevance of [Ei’s truth in vj] to [X’s

truth in vj].”

Obviously, SR entails SJC.22

This means that we can argue for the latter by

defending the former. But how can we defend SR? I believe that Gaifman’s discussion of

22 Let X be any tensed proposition and Ei&Vj<i,j>∈K⊆I×J be the agent’s time-observation partition at tn+1.

Suppose that Cn+1(X/Ei&Vj)=Cn(X in vj/Ei in vj) for any <i, j>∈K, where the truth-values of X and Ei are

38383838

Expert Principles provides a clue to this question. Here is the schema of his so-called

Expert Principles:

(Expert) C(X/pr(X)=r)=r, for all r such that C(pr(X)=r)>0.

Here, C is an agent’s credence function and pr is the agent’s “expert probability

function.” What is an expert probability function? Roughly speaking, it is a probability

function that, once it is known to the agent, will be adopted by her as her own credence

function. For instance, if you consider a local weather forecaster to be an expert for your

local weather, then C(rain/pr(rain)=r)=r where C is your credence function and pr is the

weather forecaster’s. Here, pr does not have to be a subjective probability function. When

pr is the objective chance function P, you get:

(Principal Principle) C0(X/P(X)=r)=r for any r such that C0(P(X)=r)>0,23

where C0 is an agent’s initial credence function. When pr is the agent’s future credence

function at the next epistemic moment, you get:

(Reflection) Cn(X/Cn+1(X)=r)=r for any r such that Cn(Cn+1(X)=r)>0.24

invariant within each vj for any i∈I and j∈J. Since the partition exhausts Ei&Vj such that Cn+1(Ei&Vj)>0,

Cn+1(∨<i,j>∈K (Ei&Vj))=1. Thus, Cn+1(X)=Cn+1(X&∨<i,j>∈K (Ei&Vj))=∑<i,j>∈KCn+1(X/Ei&Vj)Cn+1(Ei&Vj)=(by

supposition) ∑<i,j>∈K Cn(X in vj/Ei in vj)Cn+1(Ei&Vj). Done. (The other direction is also provable under a few

plausible assumptions but nothing in this paper hangs upon that direction of the equivalence. Still, the

equivalence is interesting because it is analogous with the equivalence between Rigidity and JC.) 23

The original version of PP is more general in that the condition can include additional information as

long as it’s admissible (Lewis 1987).

39393939

Why will an agent’s future credence function be her expert function? If the given agent is

not forgetful, she usually will be more knowledgeable in the future than in the present.

(This is a good way to see why the Reflection Principle fails in the counterexample of

Talbott (1991), in which the agent is forgetful.) By contrast, since an agent in the past is

typically less knowledgeable than the same agent in the present, an agent’s past credence

function can’t be her present expert function.

So far, so good. Now, consider this example.25

Example 7. An investor consults

a very trustworthy stock market expert. The problem is that the investor cannot reveal to

the expert some insider information she has that a company will release a new product

next month. The expert’s opinion is generally unfavorable to that company, but his

opinion conditioned upon the insider information is quite favorable. In that case, it will be

rational for the investor to make a judgment on the basis of the expert’s conditional (on

the insider information) opinion. In this situation, it seems to be rational for the investor

to adopt the expert’s conditional credence function on the product release information as

her own credence function. This intuition can be generalized as follows:

(Conditional Expert) C(X/E&pr(X/E)=r)=r, for all r such that

C(E&pr(X/E)=r)>0.

24 The original version of Reflection is more general in that the agent’s future credence function can be

from a farther future than tn+1 (van Fraassen 1984). 25

Hall discusses the same idea (Hall 2004, p. 100).

40404040

Here, C is an agent’s credence function and pr is the agent’s conditional expert

probability function on E. By “conditional expert probability function on E,” I mean a

probability function pr such that, once the function pr(-/E) and the truth of E are known

to the agent, the agent will adopt it as her credence function. Given this definition, the

Conditional Expert Principle is a natural expansion of the Expert Principle.

The Conditional Expert Principle provides a new way of understanding Rigidity.

I suggest a sub-principle of Conditional Expert below:

(Backward Reflection) Cn+1(X/Ei&Cn(X/Ei)=r)=r, for all r such that

Cn+1(Ei&Cn(X/Ei)=r)>0, where Ei is a member of the agent’s observation

partition at tn+1.

To the agent at tn+1, Ct is her previous credence function at tn. I suggest that it also must

be the agent’s conditional expert probability function at tn+1 on Ei. Why? An agent

usually has no choice but to depend on her previous credence to form the present one;

however, she is also aware that her previous credence distribution was built without her

present experience. Thus, an agent’s relation to her past credence distribution is similar to

the investor’s relation to the stock market expert in Example 7: Due to the informational

impoverishment of the agent’s previous self, it may be irrational that her present credence

in X is r given that her previous credence in X was r. Still, it is rational that her present

credence in X is r given that Ei is the true member of her present observation partition,

and her previous credence in X given Ei was r. This is because if Ei is true, then her

previous credence function conditional on Ei was a judgment made on the basis of all

41414141

information that she previously had plus the member of her observation partition actually

confirmed by her present experience.

Therefore, let’s make a plausible conjecture that Backward Reflection is usually

true of a rational agent. If we assume that the given agent correctly knows her credence

function and later remembers it with perfect confidence, Rigidity will obviously follow

from Backward Reflection.26

This provides a new way of understanding why we should

accept Rigidity.

The last step in our expansion of Gaifman’s principle is to apply the idea to

tensed propositions, especially concerning Backward Reflection. However, I suggest that

we need substantial modification to do so. Why? Consider this example. Example 8.

Again, let R be that it rains today in Boston and let P be that some form of precipitation

occurs today in Boston. In this example, an agent B at 9 AM on Monday (hereafter: tm),

knowing that it’s Monday, regards herself at 9 PM on Sunday (hereafter: ts) as an expert

about local weather in Boston except that she didn’t know whether there would be

precipitation on Monday. At tm, B learns that (i) P is true. At ts, (ii′) B’s credence in R

given P was 0.1, but (iii) her credence in [R’s truth on Monday] given [P’s truth on

Monday] was 0.3. I make two claims: First, it is not the case that B’s rational credence at

tm in R is 0.1 given (i) and (ii′), but, second, her rational credence at tm in R is 0.3 given (i)

and (iii).

26 Let r be Cn(X/E). If the agent remembers her past credence distribution with perfect confidence,

Cn+1(Cn(X/E)=r)=1. Thus, Cn+1(X/E)=Cn+1(X/E&Cn(X/E)=r)=(by Backward Reflection)r=Cn(X/E). Done.

42424242

To understand why, look at Figure 5:

Figure 5: Rain and Precipitation. The tensed proposition at the arrow tail is true on the

day of the tail’s column iff the event at the arrow head occurs on the day of the head’s

column.

On the one hand, B’s opinion, represented by the conditional credence at ts in R given P

seems to be relevant only to whether raining happens on Sunday given that a form of

precipitation occurs on Sunday. Hence, it is irrelevant to whether raining happens on

Monday given that a form of precipitation occurs on Monday, which B’s credence at tm in

R is factually about. This suggests that B’s rational credence at tm in R is not necessarily

0.1 given (i) and (ii′). On the other hand, B’s opinion at ts, represented by the conditional

credence at ts in [R’s truth on Monday] given [P’s truth on Monday], is factually all about

the italicized matter. This suggests that her rational credence at tm in R is 0.3 given (i) and

(iii).

What if B doesn’t know at tm that it is Monday? I think the natural expansion of

the above discussion is to add the condition that it is Monday. Hence, on the condition

43434343

that (i) a form of precipitation occurs now, (ii) it is Monday, and (iii) B’s past credence

function at ts was such that Cts(R on Monday /P on Monday)=0.3, the rational credence in

raining is 0.3 according to the idea of the Conditional Expert Principle. Formally,

Ctm(R/P&Vm&Ct

s(R in vm/P in vm)= 0.3)=0.3, where vm is Monday, and so Vm is that it is

Monday.

We can extract a general idea from this example. On the condition that (i)

observation proposition Ei is presently true, (ii) it is vj, and (iii) the previous credence was

such that Cn(X in vj/Ei in vj)=r where the truth-values of X and Ei are invariant within each

vj, the credence of X must be r. More formally,

(Shifted Backward Reflection) Cn+1(X/Ei&Vj&Cn(X in vj/Ei in vj)=r)=r for all x

such that Cn+1(Ei&Vj&Cn(X in vj/Ei in vj)=r)>0, where Ei&Vj is a member

of the time-observation partition at tn+1.

Of course, Shifted Rigidity follows from Shifted Backward Reflection if the agent

remembers her past credence distributions with perfect confidence.27

Hence, the idea of SR is best understood when we stipulate that from an agent’s

present point of view, the agent herself at the previous moment is an expert only lacking

the information confirmed by her present experience. For if that stipulation is true, it will

be rational for the agent to coordinate her present credence distribution with her previous

27 Let r be Cn(X in vj/Ei in vj). If the agent remembers her past credence function with perfect confidence,

Cn+1(Cn(X in vj/Ei in vj)=r)=1. Then, Cn+1(X/Ei&Vj)=Cn+1(X/Ei&Vj&Cn(X in vj/Ei in vj)=r)=(by Shifted

Backward Reflection) r=Cn(X in vj/Ei in vj). Done.

44444444

one according to SR. Since the stipulation seems to be true and SR entails SJC, we have

an argument for SJC.

F. Sleeping Beauty and Shifted Jeffrey Conditionalization

What answer does SJC give to the SB problem? In this section, I make three claims: First,

it is disprovable under SJC that SB’s credence in H stays the same from Sunday to

Monday. Second, it is provable under SJC that her credences in H1 and T1 are the same

on Monday. Third, it is not obviously provable or dis-provable that her credences in T1

and T2 are the same on Monday.

In my discussion in this section, the target tensed propositions will be H, T, H1,

T1, and T2, the evidence will be W, and the partition of intervals will be Monday,

Tuesday. The truth-values of these target tensed propositions and evidence are invariant

within Monday and within Tuesday. Hence, we can apply SJC to SB’s updating from

Sunday to Monday with the time-observation partition W&MON, W&TUE.28

Equipped with SJC, I criticize the Halfers’ thesis (1), which asserts that with no

relevant new evidence, no one can rationally change her credence in a genuine

proposition. I show that the SB problem is a counterexample of this thesis. Look at this

instance of SJC:

28 Why do we use SJC with exactly this fine-grained time-observation partition? On the one hand, the truth-

value of W is not invariant within Monday+Tuesday (0:00 AM on Monday to 11:59 PM on Tuesday).

Hence, you cannot use SJC with the time-observation partition W&(MON∨TUE). On the other hand, the

truth-value of W is invariant within each of Monday AM, Monday PM, Tuesday AM, Tuesday PM.

Hence, you can use SJC with the time-observation partition W&MONAM, W&MONPM, W&TUEAM,

W&TUEPM, but it will generate the same result as using SJC with W&MON, W&TUE. In sum, it

violates SJC’s proviso to use a more coarse-grained time-observation partition, and it is a waste of

calculating effort to use a more fine-grained time-observation partition.

45454545

(8) Cm(H) = Cs(H on Monday/W on Monday)*Cm(W&MON)+

Cs(H on Tuesday/W on Tuesday)*Cm(W&TUE).29

On the one hand, the first conditional credence is 1/2, which is equal to her credence on

Sunday in H. For “W on Monday” in the first conditional credence phrase is redundant

because she fully expected on Sunday night that she would wake up on Monday;

furthermore, “on Monday” in the resulting unconditional credence phrase also would be

redundant because H is a genuine proposition whose truth-value is insensitive to time. On

the other hand, the second conditional credence is 0. For waking up on Tuesday means

that the coin lands tails. In sum, her credence on Monday in H is the weighted average of

1/2 and 0, where the weights are her credences on Monday in W&MON and in W&TUE.

Since SB cannot rationally rule out either possibility, 0<Cm(H)<1/2. Because W doesn’t

seem to be new evidence relevant to H, this is a counterexample of (1).

In general, this explains how (1) can be violated by an SJC-er: Even when an

agent’s total evidence E is not new and relevant to X, she may change her credence in X.

For if she is uncertain whether her temporal location is in v1 or in v2, she is uncertain

between two time-observation propositions, E&V1 and E&V2. Even if the certain

evidence, E, is not new in that she fully expected that E would be true, one or both of

E&V1 and E&V2 can be new in that she didn’t fully expect E’s truth in v1 and/or in v2. In

such a case, an SJC-er can change her credence in X because arithmetically, it’s the

newness/oldness of the time-observation propositions, not of the evidence, that decides

29 In this section, I will use “… on Monday” as the abbreviation of “… is true (at every moment) on

Monday”; likewise for “… on Tuesday.”

46464646

whether it is rational for the agent to change her credence in X. (This point generalizes to

any n-case.)

Second, I defend the Thirders’ thesis (2): When SB wakes up on Monday, her

credence in H1 is the same as that in T1. As I discussed in Section A, Thirders have

defended this thesis with the intuition that SB’s credences in H and T would be equally

1/2 if she received additional information that it is Monday; hence, the actual conditional

credences in H and T, given MON, also should be 1/2. Arithmetically, this leads to thesis

(2): Cm(H1)=Cm(T1). I consider this to be a sound argument.

However, it would be more convincing if they could derive SB’s credence

distribution on Monday from that on Sunday, rather than from her credence distribution

in a counterfactual situation. With SJC, that derivation is possible. Here are its two

instances for SB’s credences at m in H1 and T:

(9) Cm(H1) = Cs(H1 on Monday/W on Monday)*Cm(W&MON)+

Cs(H1 on Tuesday/W on Tuesday)*Cm(W&TUE).

(10) Cm(T1) = Cs(T1 on Monday/W on Monday)*Cm(W&MON)+

Cs(T1 on Tuesday/W on Tuesday)*Cm(W&TUE).

According to (9), the credence on Monday in H1 is the weighted average of the

conditional credences on Sunday in [H1’s truth on Monday] given [W’s truth on Monday]

and in [H1’s truth on Tuesday] given [W’s truth on Tuesday]. The first conditional

credence is 1/2. For [H1’s truth on Monday] is equivalent to H’s truth simpliciter, and SB

fully expected W to be true on Monday; hence, it equals the credence on Sunday in H, 1/2.

47474747

The second conditional credence is 0. For H1 cannot be true on Tuesday. Hence, SB’s

conditional credence on Monday in H1 is the average of 1/2 and 0, the former weighted

by the credence in W&MON and the latter by that in W&TUE. As a result, Cm(H1)=1/2

Cm(W&MON). Similarly, it follows from (10) that Cm(T1)=1/2Cm(W&MON). Therefore,

Cm(H1)=Cm(T1).

We already established the truth of the Lesser view. However, thesis (2) provides

an alternative proof: Remember that H1, T1, and T2 exclusively exhaust all possibilities

open to SB at the moment of wakeup on Monday. She cannot rule out T2 and so

Cm(H1)+Cm(T1)<1. Since Cm(H1)=Cm(T1), Cm(H1)<1/2. Because H1 is the only possibility

in which the coin lands heads, Cm(H)<1/2.

Given this, what is the precise value of SB’s credence in H when she wakes up?

As discussed in Section A, the answer will be 1/3 if thesis (3) is true: When SB wakes up

on Monday, her credence in T1 is the same as that in T2. Unfortunately, there is no

obvious way to prove or disprove this thesis by SJC. (Try it.)

Elga says that we can prove (3) by a principle of indifference:

…even a highly restricted principle of indifference yields that you ought then to have equal

credence in each. (Elga 2000, 144.)

No doubt, “each” refers to each of T1 and T2. Thus, Elga is arguing here that (3) follows

from a highly restricted principle of indifference. In response to this, I ask two questions:

First, does (3) really follow from his principle of indifference? Second, is his principle of

indifference possibly true?

48484848

Both questions are hard to answer because Elga did not provide an explicit

formulation of his principle of indifference in his paper (2000). Fortunately, he provided

a formulation of that principle in a more recent paper (Elga 2004, 387): Define a centered

world to be a maximally consistent centered proposition.

(INDIFFERENCE) For any centered worlds X and Y, a rational agent ought to assign the

same credence to X and Y if (i) they are associated with the same possible world (i.e. for

some possible world W, X and Y both imply that W is the actual world) and (ii) they

represent epistemic situations that are subjectively indistinguishable (i.e. whichever of X

and Y is true of you, your experience will be exactly the same).

At first glance, this principle appears to entail (3): Assume that SB fully knows

everything about her world except whether the coin lands heads or tails. Under this

assumption, we can think as if T1 and T2 are centered worlds satisfying (i) and (ii). Hence,

it follows that Cm(T1)=Cm(T2). However, Weatherson (2005) criticizes this approach: First,

he says, we cannot infer (3) from INDIFFERENCE without the above assumption. For

define S to be the set of possible worlds such that for each W in S, the actuality of W is

compatible with T1 and T2, and consider the case where S is uncountably large. In this

case, it is perfectly coherent for SB to assign the same credence to any two centered

worlds associated with the same possible world in S but different credences to T1 and T2.

(For more details, see Weatherson (2005, pp. 615-616).)

Second, INDIFFERENCE is incompatible with Countable Additivity. For

assume a possible world W containing an infinite but countable number of agents who are

49494949

in the epistemic situations that are subjectively indistinguishable; if Countable Additivity

is true, it is incoherent to assign the same credence to the centered worlds representing

these agents’ epistemic situations in W. (For more details, see Weatherson (2005, pp.

619-621).) In sum, INDIFFERENCE entails (3), but it does so only under a highly

unlikely assumption, and INDIFFERENCE is incompatible with the widely accepted

axiom of Countable Additivity. Hence, the validity and factual correctness of Elga’s

argument are both questionable. That said, I leave it as an open question whether there

exists a consistent principle of indifference entailing (3).30

So far, I have disproved (1) and proved (2) by SJC. I have pointed out that (3) is

not obviously provable or disprovable under SJC. Consequently, I reject the Halfer view

and accept the Lesser view. However, I leave it as an open question whether the exact

value of SB’s credence in H is fixed by a principle of indifference or any other

consideration.

G. Conclusion

In this chapter, I have defended the Lesser view of the SB problem by SJC, a new

principle for updating de nunc credences. My discussion not only defends the Lesser

view, but it also provides a clue for what has gone wrong with the Halfer view. Read

Elga’s following comment on SB’s increasing credence in H:

30 However, I am skeptical. First, Elga points out that INDIFFERENCE leads to the weird result of “a brain

race,” although he bites the bullet (Elga 2004, p. 394). Second, Weatherson argues (convincingly in my

opinion) that INDIFFERENCE is not motivated because the intuition behind it is better captured in the

framework of imprecise credences (Weatherson 2005, p. 624). In my opinion, any principle similar to

INDIFFERENCE is likely to share these problems.

50505050

This belief change is unusual. It is not the result of your receiving new information. … So what

justifies it? The answer is that you have gone from a situation in which you count your own

temporal location as irrelevant to the truth of H, to one in which you count your own temporal

location as relevant to the truth of H. (Elga 2000, 146)

SJC provides a good explanation as to why Elga’s statement is correct: It follows from

SJC that Cm(H)=Cs(H/W on Monday)Cm(W&MON)+Cs(H/W on Tuesday)Cm(W&TUE)=

1/2Cm(MON)+0Cm(TUE)∈(0,1/2). This decrease was possible because, given that SB is

awake, TUE confirms [W’s truth on Tuesday], which is negatively relevant to the truth of

H. Since SB’s earlier knowledge that it was Sunday was not relevant to H in this way, she

has gone from [a situation in which she counts her temporal location as irrelevant] to [one

in which she counts it as relevant]. In this case, even if the evidence is old and irrelevant

to X, it is not sufficient for the rationality of not changing the credence in X.

Since the debate between Halfers and Thirders has been due not to the lack of

supporting arguments but to the failure of each side to point out the other side’s problem,

this is a good achievement. However, the even greater accomplishment seems to be the

updating principle itself. David Lewis wrote that the rule for updating de se credences is

formally identical to the rule for updating de dicto credences.31

This is wrong. As I have

demonstrated in this chapter, SJC is a good candidate for a rational principle for updating

the narrower category of de nunc credences. Obviously, SJC needs further

31 Lewis (1979) writes: “Then it is interesting to ask what happens to decision theory if we take all attitudes

as de se. Answer: very little. We replace the space of worlds by the space of centered worlds … All else is

just as before.” (Lewis 1979, pp. 533-4).

51515151

generalizations, but at least it initiates a good starting point from which we can find the

universal rule for updating de se credences.

52525252

CHAPTER III

UPDATING WITH A SEQUENCE OF OBSERVATIONS

A. Introduction

In Chapter II, I argued that a rational agent updates her credence in a tensed proposition

by SJC. In formulating SJC, I introduced this proviso: In updating from a past credence

function Cn at tn to the present credence function Cn+1 at tn+1, the agent observes nothing

between tn and tn+1.

Due to this proviso, SJC is not versatile. Compare it with SC, the traditional rule

for de dicto updating. According to the de dicto version of SC, if an agent observes E1, E

2,

E3, …, E

m during (tn,tn+m], she can simply conditionalize upon the conjunction E

1&E

2&

E3&…&E

m to update her credence in any proposition X (where X, E

1, E

2, E

3, …, E

m are

genuine propositions). This fact suggests that as a result of observing E1, E

2, E

3, …, E

m,

she learns E1&E

2&E

3&…&E

m.

Unfortunately, the same cannot be said for de se observations. Suppose that Jane

watches a cloudy sky first and a clear sky later, from 4PM to 5PM. What does she learn

as a result? If it is the conjunction of watching a cloudy sky and watching a clear sky,

then she must have learned the same thing as what she would have learned if she had

watched a clear sky first and a cloudy sky later during that period. Of course, this is

absurd. In general, when a given agent’s observations are de se, their order is important

in determining what the agent has learned as a result of those observations. Therefore,

what she learns cannot be just the conjunction of those observations.

53535353

For this reason, I chose to apply SJC only under the proviso that the agent

observes nothing during (tn,tn+1). When that proviso is satisfied, what she learns as the

result of her observations during (tn,tn+1] is simply what she observes at tn+1. In that way,

we did not have to worry about the order of observations.

However, this proviso is too restrictive. For example, think about this version of

the SB problem: SB problem 1. On Sunday, SB knows that she will experience the

following experiment. One minute later, she is put to sleep by a group of evil

experimenters. Then, they toss a fair coin. Case 1: (H) The coin lands on heads. In this

case, they awaken her only once on Monday. Case 2: (T) The coin lands on tails. In this

case, they awaken her twice, the first time on Monday and the second time on Tuesday;

between the two awakenings, they inject her with a drug that erases her memory of the

first awakening. In either case, one minute after she wakes up on Monday, she is told that

it is Monday. The experiment ends on Wednesday when she wakes up with the memory

of the previous awakening.

Let s be SB’s last conscious moment on Sunday, m be the moment of her wakeup

on Monday, and m+ be that of her being told that it is Monday. During (s,m+), she

observes W (“SB wakes up with the memory of Sunday as the last memory”). Since this

violates the proviso, SJC does not apply to SB’s credal transition from s to m+. One may

say that this is not a big problem because we can apply SJC to her credal transition from s

to m and then to her credal transition from m to m+. However, it is clear that it would be

better if we had an updating rule free from this kind of restriction.

In this chapter, I will suggest that if an agent makes a sequence of de nunc

observations E1, E

2, E

3, …, E

m during (tn,tn+m], she may update her de nunc credence in a

54545454

tensed proposition X by using a new updating rule that I call “Sequential Shifted Jeffrey

Conditionalization” (hereafter: SSJC). I will defend SSJC to some extent, but I will also

point out that SSJC does not apply to every case. Hence, I will try to provide some

criterion to distinguish the cases in which SSJC is a rational updating strategy from those

in which it is not.

B. Review of SC and SJC

In this section, I will review [the de se version of SC] and SJC. In particular, I will

explain how the latter solves a problem of the former.

Consider this case: Example 1. On Monday, Jane listens to the radio news,

which reports that today is Monday and that if there is any form of precipitation in

Boston on Tuesday, it will be rain. Consequently, Jane is sure to the degree of 0.9 that

today is Monday, but she does not completely trust the news, and so she assigns the

credence of 0.1 to the possibility that today is Tuesday. Conditional on its being Tuesday,

she ascribes no authority to the news. Hence, (i) she believes to the degree of 0.5 that [it

rains in Boston today] given that [Today is Tuesday and there is a form of precipitation in

Boston today].32

Conditional on its being Monday, Jane regards the weather news as

authoritative. Since she is pretty sure that it’s Monday, she ascribes some authority to the

news, even unconditionally. Thus, (ii) she believes to the degree of 0.8 that [it rains in

Boston on Tuesday] given that [there is a form of precipitation in Boston on Tuesday].

After listening to the news, she is put to sleep and stays in that state until next morning.

Waking up on Tuesday, Jane is told by her guru that today is Tuesday and that there is a

32 Let’s suppose that she knows that if any form of precipitation in Boston on Tuesday, it will be a rain or

snow, and she will be neutral between the two possibilities conditional on any form of precipitation there

on Tuesday.

55555555

form of precipitation today. At that moment, how confident will Jane be that it rains

today?

If Jane is a de se SC-er, she will be confident to the degree of 0.5 that it rains

today. If Jane is an SJC-er, she will be confident to the degree of 0.8 that it rains today.

Why is the former view wrong? To answer, let me formulate the de se version of SC

again: For simplicity, suppose that the agent B makes no observation during (tn,tn+1). Let

X be a centered proposition and E be the de se evidence that the agent B receives at tn+1.

Then,

(1) Cn+1(X)=Cn(X/E)

where Cn and Cn+1 are B’s credence functions at tn and tn+1. The goal from B’s point of

view at tn+1 is to find the rational credence that X is true now given the evidence that is

true now, where “now” refers to tn+1. However, the right-hand side of (1) denotes the

agent’s earlier credence that X was true then given that E was true then, where “then”

refers to tn. In this sense, if B is a de se SC-er, she comes to set her present credence in X

by consulting an outdated conditional credence. I will call this problem “the outdated

conditional credence problem.” Note: this was not a problem for the de dicto version of

SC because the target proposition and evidence have fixed truth-values in the de dicto

framework.

Consider Example 1 again. For brevity, let R be the tensed proposition that it

rains in Boston today, TUE be the tensed proposition that today is Tuesday, and P be the

tensed proposition that there is a form of precipitation in Boston today. TUE&P is Jane’s

56565656

total evidence on Tuesday. If she is a de se SC-er, her credence on Tuesday in R will be

her credence on Monday in R given TUE&P. From Jane’s point of view on Tuesday,

when she judges how likely it is that R is true now, she is consulting her credence in R’s

then truth given TUE&P’s then truth, where “now” refers to Tuesday and “then” refers to

Monday. Intuitively, this means that she comes to set her credence in R by consulting a

temporally mismatching conditional credence.

Fortunately, SJC helps us overcome the outdated conditional credence problem.

For simplicity, focus upon its sub-principle SSC: Suppose that the agent B receives no

evidence during (tn,tn+1) and knows that the time is v. Let X be a tensed proposition and E

be the de nunc evidence that the agent B receives at tn+1 (where the truth-values of X and

E are invariant within v). Then,

(2) Cn+1(X)=Cn(X in v/E in v)

where Cn and Cn+1 are B’s credence functions at tn and tn+1. In words, B sets her present

credence in X to be her previous credence that [X is true in v] given that [E is true in v].

This is intuitively reasonable, because from B’s point of view at tn+1, X is true iff “X is

true in v” is true at whatever time, and E is true iff “E is true in v” is true at whatever time.

(More about this point below.)

In Example 1, if Jane is an SSC-er, her credence on Tuesday in R will be 0.8.

For her credence on Tuesday in R will be her credence on Monday in [R’s truth on

Tuesday] given [TUE&P’s truth on Tuesday]. From her point of view on Tuesday,

57575757

(3) R is true now iff [R’s truth on Tuesday] holds at whatever time, and

(4) TUE&P is true now iff [TUE&P’s truth on Tuesday] holds at whatever

time.

These facts have two consequences: First, given these equivalences, it seems okay on

Tuesday for Jane to evaluate the credal impact of TUE&P on R by evaluating that of

[TUE&P’s truth on Tuesday] on [R’s truth on Tuesday].33

Second, there is no problem of

temporal mismatch in setting her credence on Tuesday in R with evidence TUE&P by

consulting her previous credence in [R’s truth on Tuesday] given [TUE&P’s truth on

Tuesday]. For the bracketed propositions are genuine propositions, which have fixed

truth-values. Therefore, this application of SSC is free from the outdated conditional

credence problem.

To generalize the above discussion, I introduce the following definition: Let X be

any tensed proposition and v be a temporal interval. Additionally, let V be the tensed

proposition that it is v now. Then,

(5) The de-indexicalization of X under V is the genuine proposition that X is

true throughout v.

Given (5), we can verbalize (2) into this claim:

33 By “credal impact,” I mean the quantity of the force of evidence or observation that increases or

decreases the agent’s credences. While this definition is not a good example of philosophical clarity, I am

not alone in using this somewhat opaque notion. See Lewis (1980; 272) for an example.

58585858

(6) B’s credence at tn+1 in X=B’s credence at tn in [the de-indexicalization of X

under V] given [the de-indexicalization of E under V].

The point of de-indexicalization here is to convert the target tensed proposition X and de

nunc evidence E into the equivalent genuine propositions X′ and E′ so that B can judge

how probable X is given evidence E by checking her previous credence in X′ given E′. In

this process, de-indexicalization allows the agent to use her previous conditional credence

to set her present credence without the outdated conditional credence problem.

As we have seen, SSC solves the outdated conditional credence problem by de-

indexicalizing the target tensed proposition and evidence. We can regard SJC as a

generalization of SSC where the agent is unsure about what observation she has made

and what time it is. The next question is, “How can we generalize the idea of de-

indexicalization for when the agent receives a sequence of evidence?”

C. Strategy

To answer the above question, I want to discuss SB’s credal transition from s to m+ as an

example. I begin by asking three questions: First, what does SB learn during (s,m+]?

Second, is there a genuine proposition equivalent to what she learns as a result of her

observations during that interval? Third, what is the correct way to update her credence in

H from s to m+ by making use of what she learns?

In answer to the first question, SB observes W at m and observes MON at m+. As

a result, she comes to fully believe at m+ that (E) W was previously true and MON is

presently true. In sum, E is what she learns as a result of her observations during (s,m+].

59595959

In answer to the second question, SB knows at m+ that it was previously Monday

and that the truth-value of W is invariant within Monday. Thus, she will accept this bi-

conditional at that moment:

(7) “W was previously true” is true now iff “W is true on Monday” is true at

whatever time.

Because she also knows at m+ that it is presently Monday and the truth-value of MON is

invariant within Monday, she will accept

(8) MON is true now iff “MON is true on Monday” is true at whatever time,

at m+. Now, let D be the genuine proposition that W is true on Monday and MON is true

on Monday. From the point of view at m+, D and E are equivalent because their

conjuncts are equivalent. Since D is a genuine proposition, there exists a genuine

proposition that is equivalent at m+ to E.

And now to the third question: what is the correct method of SB’s updating her

credence in H from s to m+? Since E is equivalent to D, she can evaluate the evidential

impact of E upon H by evaluating that of D on H. Hence,

(9) Cm+(H)=Cs(H/D)=Cs(H/MON is true on Monday&W is true on Monday).34

34 Since SB is sure of E at m+, Cm+(H)= Cm+(H/E). Thus, if E’s impact at m+ on H can be measured by

checking D’s impact at s on H, then (10) is derived.

60606060

It is trivial that MON is true on Monday, and on Sunday SB fully expected to wake up on

Monday. Hence,

(10) Cm+(H)=Cs(H)=1/2.

This result complies with the traditional Thirder view (Elga 2000).

Let us suppose that (9) is the correct way for SB’s updating from s to m+. If we

can generalize it, perhaps we can find a model for an agent’s updating from tn to tn+m

where the agent makes observations many times during (tn,tn+m]. The remaining question

is “How?”

In the rest of this chapter, I will proceed in the following order: In Section D, I

will present two new updating principles. In Section E, I will construct an argument for

these new principles by utilizing the idea of the Conditional Expert Principle, as I did in

Chapter II. In Section F, however, I will argue that those updating principles lead to

mutually inconsistent results when applied to the SB problem. In Section G, I will

provide a diagnosis for the problem. In Section H, I will formulate weaker versions of the

two principles suggested in Section D.

D. Updating with a Sequence of Observations

In this section, I will discuss how to update de nunc credences after making a sequence of

observations. First, I will expand our formal language L into a larger language L′, and

then I will introduce a few new definitions. Next, I will present two de nunc updating

61616161

principles by using the new language and definitions. Finally, I will illustrate how these

principles work in several examples.

I begin by expanding language L, which I constructed in the last chapter, into a

new language L′:

First, every expression in L is also a legitimate expression in L′.

Second, L′ includes an indexical “prev.” Remember the assumption that I made

in the previous chapter (see footnote 14): The agent makes observations only at a

countable number of times, say, ... tn-2, tn-1, tn, tn+1, tn+2, tn+3, .... Let us call those moments

“epistemic moments.” From now on, I will use the numeric subscripts to indicate the

order and contiguity of the epistemic moments. In other words, for any n,m∈N, tn is an

earlier epistemic moment than tm iff n<m and tn is the last moment the agent observes

anything before tn+1. Then, here is the meaning schema for “prev”:

(11) At tn, “prev” refers to tn-1.

English has no precise counterpart to “prev” in L′, but I will often use “the previous

moment” to mean the same thing.

Third, L′ includes an indexical “pres.” Here is the meaning schema:

(12) At tn, “pres” refers to tn.

Hence, “pres” in L′ amounts to “the present moment” in English.

62626262

Fourth, L′ includes (somewhat artificial) indexicals “prevk,” where k≥0. Here is

the meaning schema for “prevk”:

(13) At tn, “prevk” refers to tn-k.

As a result, “prev0” refers to the same epistemic moment as “pres” does, and “prev1”

refers to the same moment as “prev” does. This finishes our expansion of L into L′.

Next, I define two important notions. In order to do so, I ask the following

questions: First, if an agent makes a sequence of observations, what does she come to

learn at the end? Second, assuming an answer to the previous question, is there a genuine

proposition equivalent to what she learns?

Focus on the first question. Let me precisify it first: Suppose that an agent B

observes E1, E

2, E

3, ... E

m, and nothing else during interval (tn,tn+m]. Then, what does B

come to learn at tn+m as the result? Consider this answer:

(14) E1&E

2&E

3&...&E

m.

Unfortunately, this suggestion is inadequate when E1, E

2, E

3,... E

m are irreducibly tensed

propositions. For (14) ignores the temporal gaps among E1, E

2, E

3,... E

m. To understand

this point, think about the following example: Example 2. Let E1 be the tensed

proposition that the sky is cloudy now and E2 be the tensed proposition that the sky is

clear now. Suppose that Jane observes E1 at 4:30 PM and E

2 at 5:00 PM during (4:00 PM,

5:00 PM], and nothing else during that period. Then, there is a serious problem if we

63636363

regard E1&E

2 as what Jane learns as the result. For, at whatever time it is evaluated,

E1&E

2 entails that the sky is clear and cloudy at the same time. Since Jane’s perfectly

normal experience cannot lead to such absurdity, E1&E

2 is not what she comes to learn as

the result of her observations during (4:00 PM, 5:00 PM].

Instead, I suggest that what an agent B learns as the result of her observations

E1,E

2,E

3,..., E

m during (tn,tn+m] is

(15) (E1 at prevm-1)&(E

2 at prevm-2)&(E

3 at prevm-3)&...&(E

m at prev0)

in L′. Why? Note that (15) is the translation to L′ of the following expression in (my

dialect of) English:

(16) E1 was true m-1 epistemic moments ago&

E2 was true m-2 epistemic moments ago&

...

Em

is true 0 epistemic moment ago (or now).

Remember that epistemic moments are the moments at which the given agent observes

anything. If B has perfect memory as is usually assumed, then B will remember that she

observed E1 m-1 epistemic moments ago, E

2 m-2 epistemic moments ago, etc. In general,

B will remember that she observed Ek m-k epistemic moments ago, for any k such that

1≤k≤m. Thus, as a result of having observed E1, E

2, E

3,... E

m, she comes to learn the

tensed proposition expressed by (15) or (16). For example, Jane will believe at 5:00 PM

64646464

that (E1) “the sky is cloudy” was true when she observed anything last time, and (E

2) “the

sky is clear” is true now. This belief is expressed in L′ by “(E1 at prev1)&(E

2 at prev0)” or

“(E1 at prev)&(E

2 at pres).” In my opinion, this is what Jane learns as the direct result of

her observations during (4:00 PM, 5:00PM].

In general, I suggest the following definition:

(17) E is the agent B’s sequential total observation during (tn,tn+m] iff

(i) B observes E1, E

2,... E

m and nothing else during (tn,tn+m], and

(ii) E=(E1 at prevm-1)&(E


m at prev0).

(Be careful: While “E” carries the increasing subscripts from 1 to m, “prev” carries the

decreasing subscripts from m-1 to 0.)

Now, let’s focus on the second question. Again, I first precisify the given

question: If (E1 at prevm-1)&(E


m at prev0) is B’s sequential total

observation during (tn,tn+m], is there a genuine proposition equivalent to that

observation?35

To answer this question, first, I expand the notion of de-indexicalization defined

in Section B: Let E be a tensed proposition and v be a temporal interval such that the

truth-value of E is invariant within v. Thus, V is the tensed proposition that it is v now.

Then,

35 When it results in no confusion, I shall mix target language, L′, with the meta-language, English. The

same applies to the rest of this dissertation.

65656565

(18) The de-indexicalization of [E at prevk] under [V at prevk ] is [E in v].

Here is the core idea: Under the hypothesis that it was v k epistemic moments ago, E was

true k epistemic moments ago iff E was true in v.36

Hence, under the temporal hypothesis

[V at prevk], the agent will judge that [E at prevk] is true iff the genuine proposition [E in

v] is true at whatever time.

Next, I expand the notion of de-indexicalization for the case where an agent

makes a sequence of observations during an interval:

(19) The sequential de-indexicalization of

(i) (E1 at prevm-1)&(E


m at prev0)

is

(ii) (E1 in v

1)&(E

2 in v

2)&...&(E

m in v

m)

under the temporal hypothesis

(iii) (V1 at prevm-1)&(V

2 at prevm-2)&...&(V

m at prev0)

where the truth-value of Ek is invariant within v

k for any k∈1,...,m.

This looks complicated but the core idea is the same as before: Under the temporal

hypothesis (iii), (i) is true now iff (ii) is true at whatever time. Given this conditional

equivalence, the agent can evaluate the credal impact of (i) by evaluating that of (ii) when

she knows that (iii) is true. For an agent’s sequential total observation is always

36 Suppose [V at prevk]. Clearly, this supposition entails that prevk ∈v. By the definition of “in,” [E in v]

implies [E at prevk]. Since we are assuming that the truth-value of E is invariant within v, [E at prevk]

implies [E in v]. Done.

66666666

equivalent to its sequential de-indexicalization under the correct temporal hypothesis

(about the relevant epistemic moments).

Having defined these notions, I am ready to formulate my first updating rule in

this chapter, “Sequential Shifted Strict Conditionalization”: Consider a sequence of

observations E1, E

2, ...E

m and a sequence of intervals v

1, v

2, ... v

m. Assume that the truth-

value of X is invariant within vm

and that of Ek is invariant within v

k for each k∈1,...,m.

Then, for any tensed proposition X,

(SSSC) Cn+m(X)=Cn(X in vm

/(E1 in v

1)&(E

2 in v

2)&...&(E

m in v

m))

if B is sure at tn+m that for each k∈1,...,m, [Ek was/is true and it was/is

vk] at the m-k epistemic moments earlier time,

where Cn and Cn+m are B’s credence functions at tn and tn+m. Less formally, we can

rewrite SSSC in this way: Let E be (E1 at prevm-1)&...&(E

m at prev0) and V be (V

1 at

prevm-1) &...&(Vm

at prev0). Then,

(SSSC) Cn+m(X)=Cn(the de-indexicalization of X under Vm

/the sequential de-

indexicalization of E under V), if B has certainly learned until tn+m that E&V is true.

In my opinion, this is a reasonable rule for de nunc updating. It is free from the outdated

conditional credence problem, as the target tensed proposition and sequential total

observation are converted to the genuine propositions whose truth-values are fixed.

67676767

To see how SSSC works, consider this example: Example 3. Let F be the tensed

proposition that sparrows are flying away from Washington, DC today, let R be that

animals are running away from Washington, DC today, and let Q be that there is an

earthquake in Washington, DC tomorrow. On Sunday, Jane’s credence in [Q’s truth on

Tuesday] given [F’s truth on Monday]&[R’s truth on Tuesday] is 0.3. On Tuesday, she

remembers that she observed sparrows flying away from Washington, DC and that it was

Monday, and she is sure that she is observing animals running away from Washington,

DC and it is Tuesday. Jane did not have any other relevant evidence from Monday to

Tuesday. In this case, to what degree should Jane believe in Q on Tuesday?

Intuitively, her credence on Tuesday in Q must be 0.3. For she believed that [Q

would be true on Tuesday] to the degree of 0.3 conditional on the assumption that [F

would be true on Monday] and [R would be true on Tuesday], and her observations on

Monday and Tuesday exactly confirm this assumption. SSSC captures this intuition, as it

is an instance of SSSC for this example that CTUE (Q)=CSUN (Q in vT/(F in vM)&(R in

vT))=0.3, where vM is Monday and vT is Tuesday and CSUN and CTUE are Jane’s credence

functions on Sunday and Tuesday.

Next, let’s think about how to generalize SSSC for the following cases: The

agent updates from her old credence function at tn to a new credence function at tn+m, but

she is unsure what observations she has made and/or what times it has been after the m

epistemic moments earlier time (which we know to be tn). In this case, SSSC does not

apply because its proviso is not satisfied. So, what is the rational way for the agent to

update her credence?

68686868

Although SSSC does not apply to such a case, it provides an important clue for

the answer: Let Eo&Voo∈O be a partition such that Eo=(E1

o at prevm-1)&...& (Em

o at

prev0) and Vo=(V1

o at prevm-1)&...&(Vm

o at prev0) for each o∈O. I consider each Eo&Vo to

represent a possible scenario of what observations an agent B has made at what moments

(hereafter: a possible observational scenario). For simplicity, let O be 1, 2, ...p. By

SSSC:

Cn+m(X) would be

Cn(X in vm

1/D1) if B were sure at tn+m that E1&V1 is true,

Cn(X in vm

2/D2) if B were sure at tn+m that E2&V2 is true,

...

Cn(X in vm

p/Dp) if B were sure at tn+m that Ep&Vp is true,

where Do=(E1

o in v1

o)&...&(E1

o in v1

o) for each o∈O i.e., each Do is the sequential de-

indexicalization of Eo under Vo. Given these facts, it is natural that B’s credence at tn+m in

X is the weighted average of values on the right-hand sides of the above equations with

the weights coming from B’s credences at tn+m in Eo&Vo.

To formalize this idea, we need to have some preliminary jobs done. Consider a

partition &1≤k≤m((Eko&V

ko) at prevm-k)o∈O such that (i) Cn+m((E

ko&V

ko) at prevm-k)>0 for

each o∈O and (ii) ∑o∈OCn+m((Eko&V

ko) at prevm-k)=1 where Cn+m is an agent B’s credence

function at tn+m. (The intended interpretation of this partition is that each member is a

candidate for the observational scenario that B goes through during (tn+m,tn+m].) I will call

any member of this partition “(B’s) sequential time-observation proposition from tn to

69696969

tn+m.” If &1≤k≤m((Eko&V

ko) at prevm-k)o∈O also satisfies the condition that for each o∈O,

Cn(&1≤k≤m(Eko in v

ko))>0, then I will call the partition “(B’s) sequential time-observation

partition from tn to tn+m.”

Now, I am ready to formulate my next updating principle, “Sequential Shifted

Jeffrey Conditionalization”: Let &1≤k≤m((Eko&V

ko) at prevm-k)o∈O be an agent B’s

sequential time observation partition from tn to tn+m. Let Cn and Cn+m be B’s credence

functions at tn and tn+m. Assume that (#) the truth-value of X is invariant within vm

o and

that of Eko is invariant within v

ko, for any k∈1,...,m and o∈O. Then,

(SSJC) Cn+m(X)=

Σo∈OCn(X in vm

o/&1≤k≤m(Eko in v

ko))Cn+m(&1≤k≤m((E

ko&V

ko) at prevm-k)),

where Cn and Cn+m are B’s credence functions at tn and tn+m. Less formally: For each o∈O,

Eo=&1≤k≤m(Eko at prevm-k) and Vo=&1≤k≤m(V

ko at prevm-k). Clearly, Eo&Voo∈O=

&1≤k≤m((Eko&V

ko) at prevm-k)o∈O. Assume that (#) is satisfied. Then,

(SSJC) Cn+m(X)=the weighted average of Cn(the de-indexicalization of X under

Vm

o/the sequential de-indexicalization Eo under Vo) with the weights

coming from Cn+m(Eo&Vo),

70707070

where Cn and Cn+m are B’s credence functions at tn and tn+m. I believe this is a natural

generalization of SSSC for when the agent is not sure of what sequence of observations

she has made at what times.

To understand how SSJC works, think about this example: Example 4. Let F be

the tensed proposition that sparrows are flying away from Washington, DC today, RFROM

be that animals are running from Washington, DC today, RTO be that animals are running

to Washington, DC today, and Q be that there is an earthquake in Washington, DC

tomorrow. On Sunday, Jane’s credence in [Q’s truth on Tuesday] given [F’s truth on

Monday] & [RFROM’s truth on Tuesday] was 0.8, and her credence in [Q’s truth on

Tuesday] given [F’s truth on Monday]&[RTO’s truth on Tuesday] was 0.4. On Monday,

she observes that sparrows are flying away from Washington, DC and is certain of what

she is observing. On Tuesday, she observes that animals are running but is uncertain

whether they are running from Washington, DC or to Washington, DC. She observes

nothing else during the two days. On either day, she knows what day it is.

As a result, she is certain on Tuesday that she previously observed F and it was

Monday, and she is also certain on Tuesday that it is presently Tuesday. However, she is

uncertain about whether she is observing RAWAY or RBACK. Indeed, her credence on

Tuesday in [F’s previous truth and RFROM’s present truth]=0.5=her credence on Tuesday

in [F’s previous truth and RTO’s present truth]. I suggest that her credence on Tuesday in

Q will be 0.6. Look at Figure 6:

71717171

Figure 6: Evidential Uncertainty in Sequential Updating. The two rows represent two

possibilities about what Jane has observed on what days, during the last two days.

In this figure, the two rows dubbed “the first scenario” and “the second scenario”

represent the observational scenarios Jane might have gone through from Monday to

Tuesday. (i) On Sunday, Jane’s credence in Q’s truth on Tuesday was 0.8 given that [F

would be true on Monday and RFROM would be true on Tuesday]. On Tuesday, if

F&MON was previously true and RFROM&TUE is presently true, it will confirm the

bracketed condition. Therefore, Jane’s rational credence on Tuesday in Q is 0.8

conditional on F&MON’s previous truth and RFROM’s present truth. (ii) Similarly, Jane’s

credence on Tuesday in Q will be 0.4 conditional on F&MON’s previous truth and RTO’s

present truth. (iii) Therefore, her rational credence on Tuesday in Q is 0.6, the average of

0.8 and 0.4. I find this line of reasoning to be intuitive.

Interestingly, SSJC supports this intuitive claim. For it is an instance of SSJC

that CTUE(Q)=CSUN(Q in vT/(F in vM)&(RFROM in vT))*CTUE((F&VM at prev1)& (RFROM&VT

at pres))+CSUN(Q in vT/(F in vM)&(RTO in vT))*CTUE((F&VM at prev1)& (RTO&VT at pres))

=0.6, where vM is Monday and vT is Tuesday and CSUN and CTUE are Jane’s credence

functions on Sunday and Tuesday.

72727272

So far, so good. In the above example, the agent is uncertain of what she has

observed. But what if she is uncertain about what times it has been when she made those

observations? Consider this case: Example 5. Keep the meanings of “F,” “RFROM,” and

“Q” the same. On Sunday, Jane’s credence in [Q’s truth on Tuesday] given [F’s truth on

Monday]&[RTO’s truth on Tuesday] is 0.8, and her credence in [Q’s truth on Wednesday]

given [F’s truth on Tuesday]& [RTO’s truth on Wednesday] is 0.4. On that night, she

takes sleeping pills, but she realizes that she might have overdosed. If she did, she will

wake up on Tuesday. (Indeed, she didn’t overdose and will wake up and observe F on

Monday and RTO on Tuesday.)

On Monday, she wakes up and observes F. Then, she is immediately put to sleep

again, without taking any sleeping pill. (So she expects to wake up on the next day; for

the same reason, when she wakes up on the next day, she knows that only one day has

passed.) Waking up on Tuesday, Jane certainly knows that she previously observed F and

is presently observing RTO. However, since she is not sure that she didn’t overdose, she is

not sure that [it was previously Monday and it is presently Tuesday]; as far as she knows,

[it might have been Tuesday at the previous moment and it might be Wednesday now].

Consequently, her credence on Tuesday that [it was previously Monday & it’s Tuesday

now]=0.5=her credence on Tuesday that [it was previously Tuesday & it’s Wednesday

now].

73737373

Figure 7: Temporal Uncertainty in Sequential Updating. The two scenarios represent two

possibilities about what observations Jane has made on what days, for the last two days.

I claim that Jane’s credence on Tuesday in Q is 0.6. Why? Look at Figure 7:

In this figure, the two pairs of rows dubbed “the first scenario” and “the second scenario”

represent observational scenarios Jane might have gone through. (i) On Sunday, Jane’s

credence in Q’s truth on Tuesday was 0.8 given that [F would be true on Monday and

RTO would be true on Tuesday]. On Tuesday, if F&MON was previously true and

RTO&TUE is presently true, it will confirm the bracketed condition. Intuitively, her

rational credence on Tuesday in Q is 0.8 given that F&MON was previously true and

RTO&TUE is presently true. (ii) Similarly, Jane’s rational credence on Tuesday in Q is 0.4

given that F&TUE was previously true and RTO&WED is presently true. (iii) Therefore,

her credence on Tuesday in Q is 0.6, the average of 0.8 and 0.4. I find this to be intuitive

reasoning.

Importantly, SSJC supports this intuitive claim, as it is an instance of SSJC that

CTUE(E)=CSUN(E in vT/(F in vM)&(RTO in vT))*CTUE((F&VM at prev1)&(RTO&VT at

pres))+CSUN(E in vW/(F in vT)&(RTO in vW))*CTUE((F&VT at prev1)&(RTO&VW at pres))=

0.6, where vM, vT, and vW are Monday, Tuesday, and Wednesday, and CSUN and CTUE are

Jane’s credence functions on Sunday and Tuesday.

74747474

Finally, I want to provide more succinct formulations of SSSC and SSJC. For,

while the earlier formulations were good for the purpose of explanation, they often will

be too bulky for other purposes. So: Let Eo&Voo∈O be B’s sequential time-observation

partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k) and Vo=&1≤k≤m(V

ko at prevm-

k).37

Given this partition, we can define these notions: (i) Eo&Voo∈O is logically self-

optimal iff for each o∈O and k∈1,...,m, the truth-value of Eko is invariant within v

ko, (ii)

Voo∈O is logically optimal for X iff for each o∈O, the truth-value of X is invariant

within vm

o, and Eo&Voo∈O is logically optimal for X iff Eo&Voo∈O is self-optimal and

Voo∈O is optimal for X. Then,

(SSJC) Cn+m(X)=Σo∈OCn(X in vm

o/Do)Cn+m(Eo&Vo) if Eo&Voo∈O is logically

optimal for X,

where Do is the sequential de-indexicalization of Eo under Vo for each o∈O. Next, let

E&V be B’s sequential time-observation partition from tn to tn+m, where E=&1≤k≤m(Ek

at prevm-k) and V=&1≤k≤m(Vk at prevm-k). Then,

(SSSC) Cn+m(X)=Cn(X in vm/D) if E&V is logically optimal for X,

37 Note that Eo&Voo∈O=&1≤k≤m(E

ko&V

ko at prevm-k)o∈O. By the definition of sequential time-observation

partition, Eo&Voo∈O is such that (i) Eo=&1≤k≤m(Eko at prevm-k) and Vo=&1≤k≤m(V

ko at prevm-k), (ii) for each

o∈O, E1

o,…, Em

o are the candidates for B’s observations at t1,...,tn+m, (iii) for each o∈O, v1

o,…, vm

o are the

candidates for B’s temporal locations at t1,...,tn+m, (iv) Cn+m(Eo&Vo)>0 for each o∈O and

∑o∈OCn+m(Eo&Vo)=1, and (v) for each o∈O, Cn+m(Do)>0, where Do=&1≤k≤m(Eko in v

ko).

75757575

where D is the sequential de-indexicalization of E under V. Clearly, these reformulations

are equivalent to the original.

In this section, I discussed two sequential de nunc updating principles, SSSC and

SSJC. Just as SSC and SJC subsume the probabilistic inferential patterns we find to be

intuitive, SSSC and SSJC also subsume intuitive probabilistic reasoning patterns. In the

next section, I will defend these new principles by a modified version of Gaifman’s

Expert Principle.

E. A Defense of SSJC

In this section, I defend an intuitive principle that I call SSR. Since SSR entails SSJC and

SSSC, this will show that SSSC and SSJC are true of the cases similar to the example

given in this section.

First, consider this principle, which I call “Shifted Sequential Rigidity”: Let

Eo&Voo∈O be B’s sequential time-observation partition from tn to tn+m, where


ko at prevm-k). Suppose that Eo&Voo∈O is

logically optimal for X. For each o∈O,

(SSR) Cn+m(X/Eo&Vo)=Cn(X in vm

o/Do),

where Do is the sequential de-indexicalization of Eo under Vo for each o∈O. Here, Eo&Vo

codifies what observations B has made in what times, and B knows that Do is equivalent

76767676

to Eo if Vo is true. It is not difficult to see that SSR entails SSJC.38

Hence, it suffices to

defend SSR.

But how do we defend it? Think about this principle, which I will call

“Sequential Shifted Backward Reflection”: Again, let Eo&Voo∈O be B’s sequential

time-observation partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k) and

Vo=&1≤k≤m(Vko at prevm-k). Then,

(SSBR) Cn+m(X/Eo&Vo&Cn(X in vm

o/Do)=r)=r if defined,

where Do is the sequential de-indexicalization of X under Vm

o. Approximately, SSBR is

the claim that it is rational for B to set her present credence in X to be the same as her

earlier credence in the de-indexicalization of X conditional on the sequential de-

indexicalization of Eo. Remember that it is standard to presuppose that the given agent

remembers her past credence functions with perfect confidence and correctness. Under

this presupposition, SSBR entails SSR.39

In sum, SSBR entails SSR, and SSR entails SSJC. Hence, it suffices to defend

SSBR. To do so, I appeal to a more general principle, which I call “the Tensed

38 Let Eo&Voo∈O be B’s sequential time-observation partition from tn to tn+m, where Eo=&1≤k≤m(E

ko at

prevm-k) and Vo=&1≤k≤m(Vko at prevm-k). Suppose that SSR is true, i.e., Cn+m(X/Eo&Vo)=Cn(X in v

mo/Do) for

each o∈O, where Do=&1≤k≤m(Eko at v

mo). Then, Cn+m(X)=Σo∈OCn+m(X&Eo&Vo)=Σo∈OCn+m(X/Eo&Vo)

Cn+m(Eo&Vo)= (by supposition) Σo∈OCn(X in vm

o/Do)Cn+m(Eo&Vo). Done.

39

Suppose that SSBR is true, i.e., Cn+m(X/Eo&Vo&Cn(X in vm

o/Do)=r)=r for any r∈[0,1] and o∈O. Let

r=Cn(X in vj/Do). By presupposition, Cn+m(Cn(X in vj/Do)=r)=1. Thus,

Cn+m(X/Eo&Vo)=Cn+m(X/Eo&Vo&Cn(X in vm

o/Do)=r)= (by supposition) r=Cn+m(Cn(X in vj/Do)=r). Done.

77777777

Conditional Expert Principle”: For any tensed propositions X and E, any genuine

propositions X′ and E′, and any temporal hypothesis V,

(TCE) C(X/E&V&prt(X′/E′)=r)=r if defined and these conditions are met:

(a) C is an agent B’s present credence function and prt is an agent Ex’s

credence function at t on which B depends to decide her present

credal opinion,

(b) Ex did not know at t whether E′ was true, B can presently access all

information that Ex had at t, and B has observed no other

information possibly except E.

(c) B presently knows that if V is true, [X is presently true iff X′ is true

at t] and [E is presently true iff E′ is true at t].

Suppose (a)-(c). By (a), it seems rational for B to have her credence in X somehow

coordinated with Ex’s credence distribution. But what is the rational way to do so?

To answer, think about the facts that follow from the following conditions

(under (a)-(c)): (i) E is presently true, (ii) V is presently true, and (iii) Ex’s credence at t in

X′ given E′ is r. It follows that E′ is true, which Ex did not know at t. Given this

informational impoverishment, B cannot rationally depend upon Ex’s unconditional

credences at t to set her present credences, conditional on (i)-(iii). Nevertheless, it is still

plausible that it is rational for B to use Ex’s credences conditioned upon E′. For whether

E is true is the only information that B presently has but that Ex might have lacked at t,

and E’s present truth is equivalent to E′’s truth at t. Hence, it will be rational for B to use

78787878

(iii) to set her present credence. Given this reasoning, it is rational that B’s present

credence in X is r on the condition that (i)-(iii) are true.

I claim that SSBR is a sub-principle of TCE. Consider the following instance of

SSBR, which involves B’s credal transition from tn to tn+m: Let Eo= &1≤k≤m(Eko at prevm-k),

Vo=&1≤k≤m(Vko at prevm-k), and Do=&1≤k≤m(E

ko in v

ko ). So Do is the sequential de-

indexicalization of Eo under Vo. Then,

(20) Cn+m(X/Eo&Vo&Cn(X in vm

o/Do)=r)=r if defined.

First, it is inevitable for B to set her present credence by depending upon her earlier

credence distribution. Second, Eo, if true, represents B’s total observation during (tn, tn+m],

which B could not access at tn. Third, if Vo (⊃Vm

o ) is true, X is true iff [X in vm

o] is true,

and Eo is true iff Do is true. Hence, B satisfies the provisos of TCE with respect to her

credence distribution at tn+m. Therefore, SSBR follows from TCE as a special case.

Consider this example: Example 6. Keep the meanings of “F,” “RFROM,” and

“Q” the same as in the earlier examples. At 9:00 AM on Tuesday (hereafter: t), Jane

regards herself at 9:00 PM on Sunday (hereafter: s) as an expert about local natural

phenomena. Then, what should her credence be at t in Q, given that (i) (E) F was

previously true and RFROM is presently true, (ii) (V) it was Monday previously and it is

Tuesday now, and (iii) her credence at s in Q’s truth on Tuesday was 0.7 given that F

would be true on Monday and RFROM would be true on Tuesday? My answer: it must be

0.7. Formally, Ct(Q/E&V&Cs(Q on Tuesday/D) =0.7)= 0.7, where D is the de-

79797979

indexicalization of E under V, i.e. the genuine proposition that F is true on Monday and

RFROM is true on Tuesday.

Why? Consider Figure 8:

Figure 8: Flying Birds, Running Animals, and Earthquake. The belief at the tail of an

arrow is true on the day of the belief’s column iff the event at the head of the arrow

occurs on the day of the event’s column.

First, to set her credence at t in Q, Jane seems to have no other choice but to use her

earlier credence distributions, possibly that at s. However, second, she must not simply

adopt her unconditional credence at s in Q as her credence at t in Q. This is because Jane

has observed some facts relevant to Q, such as birds flying away and animals running

away, after s. So what is the rational way for Jane to set her credence at t in Q by utilizing

her credence distribution at s? Third, here is a suggestion: Let E be (F at prev)&(R at pres)

and V be (MON at prev)&(TUE at pres). Then, Jane’s credence at t in Q is r on the

assumption that (i) E is presently true, (ii) V is presently true, and (iii) her credence at s in

80808080

[Q on Tuesday] is r given D, where D=(F on Monday)&(RFROM on Tuesday). For

although Jane cannot trust her credal judgments on Sunday for the reason mentioned, she

can still trust those conditioned upon D, which codifies her observations after Sunday.

Plus, Jane knows on Tuesday that if V is true, then Q is equivalent to [Q on Tuesday],

and so she can rationally set her credence in Q by checking her credence on Sunday in [Q

on Tuesday] conditioned upon D. (We need to substitute “must” for “can” in the last

sentence in light of the outdated conditional credence problem, which we discussed in

Section B.)

We can generalize this reasoning: Cn+m(X/Eo&Vo&Cn(X in vm

o/Do)=r)=r where

Eo=&1≤k≤m(Eko at prevm-k) is possibly the agent B’s sequential total observation during

(tn,tn+m], Vo=&1≤k≤m(Vko at prevm-k) is a temporal hypothesis about when B has made

observations during (tn,tn+m], and Do=&1≤k≤m(Eko in v

ko) is the sequential de-

indexicalization of Eo under Vo. It is easy to see that at tn+m, [X in vm

o] and Do are genuine

propositions equivalent to X and Do respectively. Hence, it is intuitively rational at tn+m

for B to defer to herself at tn in the suggested way. It follows that SSBR is true.

So far, I have argued that (i) since an agent at tn+m will usually regard herself at tn

as an expert only lacking the information of what sequence of observations she would

make, SSBR is the correct method of deference to her past self, (ii) under the usual

presupposition of perfect memory, SSR follows from SSBR, and (iii) since SSR entails

SSJC, SSJC and SSSC are the correct rules for de nunc updating.

However, I suspect that the reasoning provided in this section doesn’t justify all

instances of SSSC or SSJC. In the next section, I will apply SSSC to the SB problem and

81818181

show that it yields an inconsistent result. In Section G, I will try to identify the source of

the problem and how to fix SSSC and SSJC.

F. The SB Problem and the Inconsistency of SSJC

In this section, we apply SSSC to the SB problem. Interestingly, it will be shown that if

SB updates her credence in H in accordance with SSSC, it leads to an inconsistent result.

Remember the temporal structure of the SB problem:

(s) SB is put to sleep on Sunday.

(m) SB wakes up on Monday.

(m+) SB is told that it is Monday.

We can think about three credal transitions: (i) the transition from s to m, (ii) that from m

to m+, and (iii) that from s to m+. There are two possible strategies for SB to update from

s to m+: (a) updating from s to m+ all at once or (b) updating from s to m and m to m+

step by step. This raises a concern: What if the results of (a) and (b) do not match?

Unfortunately, this concern is legitimate. First, consider this instance of SSSC,

for the updating from m to m+:

(21) Cm+(H)=Cm(H on Monday/MON on Monday)=Cm(H)<1/2.

This is a correct instance of SSSC because SB is sure at m+ of MON&MON (the

conjunction of her observation proposition and the interval information, which happen to

both be MON in this case). First, [MON on Monday] is redundant because “it’s Monday”

82828282

is trivially true on Monday. Second, [H on Monday] is equivalent to H because H is a

genuine proposition whose truth-value is insensitive to time. Thus, Cm+(H)=Cm(H).

However, in the previous chapter, we already determined that Cm(H)<1/2.

Next, consider this instance of SSSC for the updating from s to m+:

(22) Cm+(H)=Cs(H on Monday/W on Monday & MON on Monday)=1/2.

This is a legitimate instance of SSSC because SB is sure at m+ that W&MON was

previously true and MON&MON is presently true. In (22), [W on Monday & MON on

Monday] is redundant because SB knew that she would wake up on Monday and “it’s

Monday” would be true on Monday. Also, [H on Monday] is just the same as H. Hence,

Cm+(H)=Cs(H)=1/2.

Obviously, (21) and (22) are mutually inconsistent. Since both are instances of

SSSC, and so of SSJC, this means that they are inconsistent updating rules.

G. A Diagnosis and a Potential Solution

So SSSC (which is a special case of SSJC) is false because it leads to mutually

inconsistent results. Does this mean that we should totally abandon SSSC and SSJC? No.

In this section, I discuss Elga’s view that temporal knowledge is an essential element of

expertise, and I argue that we have to modify SSSC and SSJC slightly so that the

modified rules do not apply to the cases where the agent suffers from temporal ignorance.

Fortunately, this modification enables us to avoid the aforementioned inconsistency.

83838383

Elga (2007) argues that a rational agent doesn’t have to obey Reflection when

she expects herself to suffer from temporal ignorance:40

There is another sort of information loss, a sort associated with losing track of … what time it is.

Information loss of that sort can also lead to violations of Reflection. For example, suppose that

you are waiting for a train. You are only 50% confident that the train will ever arrive, but you are

sure that if it does arrive, it will arrive in exactly one hour. Since you have no watch, when fifty-

five minutes have in fact elapsed you will be unsure whether an hour has elapsed. As a result, at

that time you will have reduced confidence—say, only 40% confidence—that the train will arrive.

So at the start, you can be sure that when fifty-five minutes have elapsed, your probability that the

train will ever arrive will have gone down to 40%. So your anticipated imperfect ability to keep

track of time creates a violation of Reflection. (Elga 2007, 482)

Let A be the proposition that the train arrives at some time, let CINIT be the agent’s

credence function at the initial moment, and let C55 MIN+ be her credence function in fifty-

five minutes. In the above example, it is an instance of Reflection that CINIT(A/C55 MIN+(

A)=0.4)=0.4, but CINIT(A)=CINIT(A/C55 MIN+(A)=0.4)=0.5. Elga claims that this violation is

understandable because Reflection is a special case of Gaifman’s Expert Principle and the

agent, at the initial time, must not regard herself in fifty-five minutes as an expert given

the expected loss of the track of what time it is.

I find Elga’s claim to be plausible. After all, the agent initially knows that in

fifty-five minutes, only fifty-five minutes will have passed since the initial moment, but,

40 Reflection is the principle that for any proposition X and any real number r, Cn(X/Cn+m(X)=r)=r if defined,

where Cn and Cn+m are agent B’s credence functions at tn and tn+m. Also see the previous chapter for its

relation to Gaifman’s Expert.

84848484

when fifty-five minutes have actually passed, the agent cannot know it because of her

temporal ignorance. Thus, there seems to be some information that she can access

initially but that she cannot access in fifty-five minutes. This relative ignorance makes it

irrational for the agent’s initial self to defer to her future self in fifty-five minutes as

recommended by Reflection.

If temporal knowledge is an essential element of expertise required by Reflection,

it is plausible that temporal knowledge is also a necessary condition of expertise required

by Shifted Backward Reflection. To see this point more clearly, think about the following

variant of the SB problem: SB problem 2. On Sunday, SB knows that she will go

through the following experiment: On that night, evil experimenters will put her to sleep

and toss a fair coin. Case 1: (H) The coin lands heads. Then, she is awakened once in a

room with a big electronic calendar. The calendar is slightly faulty because it has a 0.2

chance of showing the day of tomorrow. Case 2: (T) The coin lands tails. In this case, SB

is awakened in the same room twice, the first time on Monday and the second time on

Tuesday. Plus, the experimenters inject her with a drug with the effect of erasing her

memory of Monday at some time between her two awakenings. In either case, one minute

after she wakes up on Monday or Tuesday, a completely reliable person tells her what

day it is.

As before, let s be the last conscious moment on Sunday, let m be the moment of

wakeup on Monday, and let m+ be the moment of being told that it’s Monday. Also, let

WTUE be the tensed proposition expressed by “SB wakes up today watching the calendar

displaying ‘TODAY IS TUESDAY.’” As a matter of fact, when she wakes up on

Monday, SB receives WTUE as evidence.

85858585

Let’s think about SB’s (i) credal transition from s to m, (ii) that from m to m+,

and (iii) that from s to m+. If we apply SSJC to (i)-(iii), what will it say about her

credences at m and m+ in H?

First, we get the following result by applying SJC (which is a special case of

SSJC) to SB’s credal transition from s to m:

(23) Cm(H)=Cs(H on Monday/WTUE on Monday)Cm(WTUE&MON)+

Cs(H on Tuesday/WTUE on Tuesday)Cm(WTUE&TUE).

Which message the calendar shows on Monday is clearly irrelevant to the result of the

coin toss, and so the first conditional credence is 1/2. Whatever the calendar displays on

Tuesday, waking up on Tuesday entails the coin’s landing tails, and so the second

conditional credence is 0. Hence,

(24) Cm(H)=1/2Cm(WTUE&MON).

Since SB is sure at m that she is waking up reading “TODAY IS TUESDAY” on the

calendar,

(25) Cm(H)=1/2Cm(MON/WTUE).

However, the calendar has a 0.2 chance of displaying the day of tomorrow. Thus, if it is

displaying “TODAY IS TUESDAY,” it has a 0.8 chance of displaying today’s date, in

86868686

which case today is of course Tuesday, but it also has a 0.2 chance of displaying

tomorrow’s date, in which case today is Monday. Intuitively, her credence at m in MON

is 0.2 given WTUE as evidence. Therefore,

(26) Cm(H)=0.1.

Second, we get the following result by applying SSC (which is a special case of

SSSC and therefore of SSJC where the length of the sequence of evidence is 1) to SB’s

credal transition from m to m+:

(27) Cm+(H)=Cm(H on Monday/MON on Monday)=Cm(H).

For SB is told at m+ that it is Monday, but it is no secret that MON is certainly true on

Monday.

Third, we get this result by applying SSSC (which is a special case of SSJC) to

SB’s credal transition from s to m+:

(28) Cm+(H)=Cs(H on Monday/MON on Monday &WTUE on Monday)

=Cs(H)=1/2.

For she knew at s that MON would be true on Monday and, whether the electronic

calendar works correctly or not, it is irrelevant to whether the coin lands on heads.

87878787

Of course, (26)-(28) are jointly inconsistent; if they are all true, SB’s credence in

H is both 0.1 and 1/2 at the same time. Thus, just as in SB problem 1, SSJC leads to

inconsistency in SB problem 2. The difference is that in the latter case, it is easier to see

which result of its application is wrong for what reason: (27) is false because SB doesn’t

know at m what time it is and SSJC doesn’t work correctly when it is used for the credal

transition from a moment of temporal ignorance.

To understand why, remember that I argued for SSJC on the basis of SSBR. For

my present purpose, it is easier to talk directly in terms of SSBR. Hence, see these

instances of SSBR for SB problem 2:

(29) Cm(H/WTUE&MON&Cs(H on Monday/WTUE on Monday)=1/2)=1/2.

(30) Cm(H/WTUE&TUE&Cs(H on Tuesday/WTUE on Tuesday)=0)=0.

(31) Cm+(H/MON&MON&Cm(H on Monday/MON on Monday)=r)=r,

where r=Cm(H).

(32) Cm+(H/[WTUE at prev & MON at pres]&[MON at prev & MON at

pres] & Cs(H on Monday/MON on Monday & WTUE on Monday)=1/2)=1/2.

First, (29) and (30) are the instances of SBR (and therefore of SSBR) for SB’s credal

transition from s to m. Under the assumption that she always remembers her past

credence functions with perfect correctness and confidence, it follows from them that

Cm(H/WTUE&MON)=1/2 and Cm(H/WTUE&TUE)=0, which eventually leads to (26).

Second, (31) is an instance of SBR for her updating from m to m+. Under the same

88888888

assumption, it leads to (27). Third, similarly, (32) leads to (28). Therefore, there is a

contradiction.

The common idea behind these instances of SSBR is that SB must consider

herself at a past moment as an expert in a limited sense.41

For while the agent has no

choice but to depend on her past self to set her present credences, she is also aware that

she has acquired some new information. Thus, I suggested that, roughly, SB must defer

not to her unconditional past credence distribution but to her past credence distribution

conditioned upon the observations that she has made after she has the past credence

distribution.

However, this idea fails to support (31). For when she is told that it is Monday,

SB realizes that it was due to the faultiness of the electronic calendar that she was

strongly biased to the possibility that it was Tuesday then. Being aware of this fact, she

shouldn’t trust her own previous credal judgment, which was based upon that faulty

temporal information.

In general, just as a rational agent cannot consider her future self to be an expert

when she expects to lose track of time in the future, a rational agent cannot regard her

past self to be an expert when she remembers that she lost track of time in the past. For

this reason, I believe that SSBR is not true of cases in which the agent previously did not

know what time it was. Since SSJC gains its plausibility from SSBR, we should apply

SSJC only to cases in which the agent knew what time it was.

Given this diagnosis, let’s think about how it affects the three credal transitions

in the original SB problem: (i) the credal transition from s to m, (ii) that from m to m+,

41 Elga uses “guru” to refer to an agent who is an expert in this limited sense. See Elga (2007).

89898989

and (iii) that from s to m+. If we apply SSJC to all these credal transitions, we are

confronted with a contradiction: By applying SJC to (i), I have shown that SB’s credence

at m in H is less than 1/2. By applying SSC to (ii), I have proven that her credence at m+

in H has the same value as her credence at m in H. However, by applying SSJC to (iii), I

have shown that her credence at m+ in H is 1/2. Hence, 1/2>Cm(H)=Cm+(H)=1/2. A

contradiction.

Fortunately, our discussion in this section suggests that it is faulty to apply SSC

to (ii). For look at Figure 9: As you see in this figure, SB didn’t know at m that it was

Monday, having lost track of what time it was. Thus, at m+, SB cannot regard herself at

m as an expert. If SSJC and its sub-principles are correct only when the agent knew what

time it was at the time from which she is updating, this means that SSC does not apply

correctly to SB’s credal transition from m to m+.

Figure 9: Deference and Temporal Ignorance. It is not necessarily the case that Cm+(H)=Cm(H)=r<1/2.

For it is irrational at m+ for SB to defer to her previous credal judgment, provided that she suffered

from temporal ignorance at m.

90909090

This suggests the following view: When she wakes up on Monday, SB’s

credence in H is lower than 1/2 by SJC applied to (i), and, when she is told that it is

Monday, her credence in H is 1/2 by SSSC applied to (iii). This is exactly the popular

thesis of Thirders.

The discussion until now suggests that we have to modify SSSC and SSJC by

adding the proviso that to update one’s credences from tn to tn+m using these rules, an

agent B must be sure at tn+m that she didn’t suffer from temporal ignorance at tn. Here are

the results of this modification, which I call “Restricted Sequential Shifted Jeffrey

Conditionalization*” and “Restricted Sequential Shifted Strict Conditionalization*”: Let

Eo&Voo∈O be B’s sequential time-observation partition from tn to tn+m, where


ko at prevm-k). Then,

(RSSJC*) Cn+m(X)=Cn(X in vm

o/Do)Cn+m(Eo&Vo) if Eo&Voo∈O is logically optimal

for X and B was free from temporal ignorance at tn,

where Do is the sequential de-indexicalization of Eo under Vo for each o∈O. Next, let

E&V be B’s sequential time-observation partition from tn to tn+m, where E=&1≤k≤m(Ek

at prevm-k) and V=&1≤k≤m(Vk at prevm-k). Then,

(RSSSC*) Cn+m(X)=Cn(X in vm/D)Cn+m(E&V) if E&V is logically optimal for X and

B was free from temporal ignorance at tn,

91919191

where D is the sequential de-indexicalization of E under V. Now, (21) and (27), the

troublesome instances of SSSC in SB problems 1&2, don’t follow from RSSSC*

because its proviso is violated.

Also note that SSC and SJC are the special cases of SSSC and SSJC, respectively,

where the length of sequential timed evidence is 1. Just as we modified the sequential

principles into their restricted versions, we can restrict SSC and SJC with the additional

proviso that the agent was previously free from temporal ignorance. Let’s call the

restricted versions of SSC and SJC “Restricted Shifted Strict Conditionalization*” and

“Restricted Shifted Jeffrey Conditionalization*”. (Since the formal modification is

obvious, I do not provide explicit formulation of RSSC* and RSJC* here.)

In this section, I have provided a diagnosis of the problem discussed in Section F

and discussed how to modify SSSC, SSJC, SSC, and SJC accordingly. The modification

seems successful in removing the aforementioned problem.

H. Conclusion

In this chapter, I have argued for two new principles for sequential updating. I tried to

defend those principles, SSSC and SSJC, by a reasoning similar to that which I used for

SSC and SJC in Chapter II.

However, SSSC and SSJC turned out to be inconsistent when applied to the three

updating paths in the SB problem. With the help of Elga’s discussion, I provided a

diagnosis for the problem of SSSC and SSJC, demonstrating that they are inconsistent

when updating from a credence function when the agent suffers from temporal ignorance.

Accordingly, I suggested weaker modifications of these principles, RSSSC* and RSSJC*.

92929292

These yet new principles are not obviously inconsistent because they don’t apply to cases

in which the agent suffers from temporal ignorance.

Seemingly, this leads to a happy ending: We have updating rules that are free

from obvious inconsistencies and that are justified by plausible arguments. Nevertheless,

I am not perfectly satisfied. On the one hand, the provisos of RSSSC* and RSSJC* are

too restrictive. For it is clearly desirable to have updating rules applicable even to a credal

transition at the initial time of which the agent did suffer from temporal ignorance. On the

other hand, those provisos may not be sufficiently restrictive. For although RSSSC* and

RSSJC* didn’t result in any apparent contradictions in the discussed counterexamples to

SSSC and SSJC, there is no guarantee that we will not find other counterexamples in

which the restricted versions also lead to contradictions.

In the succeeding chapters, I will pursue the following goals: First, I will

formulate and defend general principles for updating de nunc credences that are

applicable to a credal transition with initial temporal ignorance. Second, I will show that

those new principles are free from the problems of SSSC and SSJC that I have discussed

in this chapter.

93939393

CHAPTER IV

UPDATING WITH DE PRIORI INFORMATION

A. Introduction

In the last chapter, I suggested a rule that a rational agent can use to update her de nunc

credences after making a sequence of observations. Now, we face a new challenge:

Suppose that you learn new information about what time it was at an earlier epistemic

moment. For example, you wake up without knowing whether today is Monday or

Tuesday and, after a while, you newly learn that it was Monday when you woke up earlier.

In such a case, what will the rational rule be for updating your credence in a tensed

proposition?

None of the rules discussed in Chapter III will help you to find the answer: First,

SSJC does not apply correctly to any such case. For you can learn new information about

what time it was at an earlier time t only when you did not know “what time it is now” at

t, and I already discussed the fact that it would be irrational for you to use the rule of

SSJC if you are updating from a moment of temporal ignorance. Second, the less general

rules discussed in the last section—SSC, SJC, and SSSC—will not apply correctly to

such a case because they are the sub-principles of SSJC.42

Due to this problem, we need yet another rule for updating. In this chapter, I will

suggest that when an agent newly learns information about what time it was at an earlier

42 Moreover, RSSJC* and its sub-principles (RSSSC*, RSSC*, and RSJC*) do not apply to such a case

because freedom from temporal ignorance is the common proviso of those rules.

94949494

epistemic moment (hereafter: de priori information), the agent ought to update her de

nunc credences by following the updating rule that I call “General Shifted Jeffrey

Conditionalization” (hereafter: GSJC). In particular, I want to achieve these goals: to

defend GSJC and then to illustrate how it works with examples. Finally, I will apply it to

SB’s credal updating from m to m+, arguing that the resulting credence at m+ in H is 1/2.

B. Strategy

In this chapter, my goal is to find a rule for updating de nunc credences. The rule should

be applicable to a credal transition from tn to tn+m even if the agent is ignorant at tn of

what time it is then. To find a clue for such a principle, consider SB’s credal transition

from m to m+. If SB updates her credences in accordance with SSC,

(1) Cm+(H)=Cm(H/MON on Monday)=Cm(H)<1/2.

Here is a rationale for this claim: When told that it is Monday, SB has no choice but to

consult her previous credal judgments to set her present credence in H. However, she

made those judgments before learning that it is Monday. Hence, SB needs to consult her

previous credal judgments conditioned upon MON or something equivalent. In a similar

case, it is generally better to consult one’s previous credal judgments conditioned upon

the de-indexicalization of one’s present observations than to consult those conditioned

upon the present observations themselves.43

Thus, her credence at m+ in H will be equal

to her credence at m in H conditioned upon [MON on Monday].

43 This is because of the so-called outdated conditional credence problem, which I discussed in the last

chapter.

95959595

In the last chapter, I criticized this rationale: The above rationale assumes that

when she is told that it is Monday, SB can regard her previous self as an expert only

lacking what she has just learned. However, her previous self cannot be relied upon as an

expert even in this limited sense. For she did not previously know what day it was then,

and such knowledge is a necessary condition even for the sort of expertise we are talking

about.

To find a better way to set her credence at m+ in H, we must remember that

when SB wakes up on Monday, she is ignorant of what time it is then, and when she is

told that it is Monday, she comes to know something which frees her from her previous

ignorance. With this in mind, I pose three questions: First, what are the contents of SB’s

ignorance and knowledge? Second, what is the logical relation between those contents?

Third, given this relation, what is the rational way for SB to update her credence in H

from m to m+?

Let’s focus on the first question. Waking up on Monday, SB is ignorant of the

fact that “it is Monday at the present moment,” and when she is told that it is Monday,

she acquires the knowledge that “it was Monday at the previous moment” by inference.

The contents of the ignorance and knowledge are those expressed by the sentences

quoted in the last sentence.

This answer is very plausible. For both “the present moment” and “the previous

moment” refer to the same moment (=m), and so it is easy to see how the later

acquirement of knowledge removes the earlier temporal ignorance. Thus we can say:

When she is told that it is Monday, SB is aware of this fact:

96969696

(2) By being ignorant of MON, I previously suffered from temporal ignorance,

from which I have been freed by coming to know [MON at prev].

In general, if an agent previously suffered from ignorance of V, she can be freed from that

ignorance by learning [V at prev] now.44

Now let’s focus on the second question. As previously mentioned, when SB

wakes up, she is ignorant of the fact that “it is presently Monday,” but when told that it is

Monday, she comes to know that “it was previously Monday.” Between the contents of

these ignorance and knowledge, the following logical relation holds: When told that it is

Monday, SB knows that

(3) [MON at prev] is presently true iff MON was previously true.

In general, [V at prev] is presently true iff V was previously true, where V is a tensed

proposition specifying what time it is.

Turning to the third question, I claim that the following equation describes a

rational way for SB to update her credence in H from m to m+:

(4) Cm+(H)=Cm+(H/MON at prev)=Cm(H/MON)=1/2.

44 It is true at m+ that (*) SB is freed from her previous ignorance of MON by coming to know MON.

However, it is not generally true that if an agent previously suffered from the ignorance of V, she can be

freed from that temporal ignorance by learning V. Consider this case: Jake goes to bed not knowing THU,

but he is freed from that ignorance by coming to know [THU at prev] the next morning. In the last sentence,

you cannot replace [THU at prev] with THU salva veritate.

97979797

Here is the rationale for this claim: When she is told that it is Monday, SB realizes that it

was previously Monday but remembers that she did not know what day it was then.

Hence, she will regard her previous self as ignorant of what day it was then. Still, she has

no choice but to depend upon her previous credal judgments to set her present credence

in H. In this situation, she will be better off consulting her previous credal judgments

conditioned upon [MON at prev] or something equivalent. According to (3), [MON at

prev] is true iff MON was true at the previous moment. So it seems rational for her to set

her present credence in H by consulting her previous credence function conditioned upon

MON.

Is this rationale for (4) vulnerable to the same criticism I raised against the

rationale for (1)? No. As (2) says, MON is the content of SB’s temporal ignorance at the

moment of her waking up. Since the consulted conditional credence function is

conditioned upon MON, it is a credal judgment based upon the temporal information that

was correct at the mentioned moment.

Hence, I believe that (4) captures the correct updating pattern for SB’s credal

transition from m to m+. The remaining job is to incorporate this pattern into a new

principle for de nunc updating. For this project, I will proceed in the following order: In

Section C, I will present a general rule for updating de nunc credences. In Sections D and

E, I will defend that rule using a yet new variant of the Conditional Expert Principle. In

Section F, I shall reformulate the thus defended rule into a more readily usable form. In

Section G, I will apply it to SB’s credal transitions from s to m, from m to m+, and from s

to m+. The results will turn out to be mutually consistent. In Section H, I will discuss the

98989898

relation between the rules presented in this chapter and those presented in the earlier

chapters, such as SSC, SJC, SSSC, and SSJC.

C. Updating with De Priori Information

I will begin this section by defining a few important notions related to information about

what time it was. Next, I will present a new principle for de nunc updating, applicable to

a credal transition from a moment of temporal ignorance, and illustrate how it works with

an example. Finally, I will provide a shorter formulation of that principle, which I will

use in later discussions.

To begin, I ask the following questions: Consider an agent B’s credal transition

from tn to tn+m. First, at tn+m, how can B specify what times it had been until an earlier

epistemic moment? Second, how can we (sort of) translate such information from the

context of tn+m to the context of tn? Third, what will the logical relation be between the

original and translated pieces of information?

To answer the first question, I introduce the following definition:

(5) Tensed proposition F is de priori information iff F is [W at prevk] for

some special tensed proposition W and some number k≥1.

For example, consider this case: Example 1. On Sunday night, Jane knows that it is

Saturday or Sunday but does not know which. The next morning, her sincere friend Jeff

tells her, “The last time you were awake it was Sunday.” From this testimony, Jane learns

[SUN at prev(1)]. By definition, it is de priori information.

99999999

This notion of de priori information can be further generalized:

(6) Tensed proposition W is sequential de priori information iff W=(W1 at

prevk)&(W2 at prevk+1)&... &(W

n at prevk+n-1).

For instance, consider this case: Example 2. Briefly waking up on Sunday night, Jane

knows that either [it is Sunday now and it was Saturday at the previous moment] or [it is

Saturday now and it was Friday at the previous moment], but she does not know which.

The next morning, Jeff tells her, “The last time you were awake it was Sunday, and when

you were awake before that moment it was Saturday.” From this testimony, she learns

(SUN at prev1)&(SAT at prev2). By definition, it is sequential de priori information. As

you can easily see, such information is about what times it had been until an earlier

epistemic moment.

Next, I introduce a definition for a special case of sequential de priori

information, namely, information about what times it had ever been until an earlier

epistemic moment: For simplicity, I suppose that B has the first epistemic moment t0, i.e.,

B received her first evidence at t0. (It is possible but less elegant to discuss my view

without this supposition.) Then,

(7) W is a temporal description at tn+m of the epistemic moments until tn iff

W=(W1 at prevm)&(W

2 at prevm+1)&…&(Wn+1

at prevm+n).

100100100100

At tn+m, “prevm+n” refers to t0. Hence, the temporal description at tn+m of the epistemic

moments until tn exhaustively specifies what times it had been at the epistemic moments

between t0 and tn. This answers the first question.

To answer the second question, I introduce this notion:

(8) [W at prevk-m] is the re-indexicalization of [W at prevk] for the m epistemic

moments earlier time.

Consider Example 1 again. When Jane learns [SUN at prev1] from Jeff, SUN (or [SUN at

prev0]) is its re-indexicalization for the previous epistemic moment. We can generalize

this notion for sequential de priori information:

(9) R is the sequential re-indexicalization of W for the m epistemic moments

earlier time iff

(i) W=(W1 at prevk)&(W

2 at prevk+1)&…&(W

n at prevk+n) and

(ii) R=(W1 at prevk-m)&(W

2 at prevk+1-m)&…&(W

n at prevk+n-m).

To understand this definition, consider Example 2 again. When Jane learns (SUN at

prev1)&(SAT at prev2) from Jeff, (SUN at prev0)&(SAT at prev1) is its sequential re-

indexicalization for the previous moment. In a good sense, (SUN at prev0)& (SAT at prev1)

is the “translation” of (SUN at prev1)&(SAT at prev2) from Monday morning to Sunday

night.

101101101101

Now, remember that the temporal description at tn+m of the epistemic moments

until tn exhaustively specifies what times it had been until tn. By definition, such a

temporal description is sequential de priori information, and so it will have a suitable

sequential re-indexicalization. Thus, I make this claim: Consider the temporal description

W at tn+m of the epistemic moments until tn. Then, W’s sequential re-indexicalization R

for the m epistemic moments earlier time will be, in a good sense, the translation of W

from tn+m to tn.

To explain why, we need to answer the third question: Let W be a temporal

description at tn+m of the epistemic moments until tn, and let R be its re-indexicalization

for the m epistemic moments earlier time. Then,

(10) W is true at tn+m iff R was true at tn.

Hence, both W and R describe what times it had been until tn, the former from the point

of view at tn+m and the latter from the point of view at tn.

I have answered all three questions, but I still need one more definition to

precisely formulate the rules I am working toward: Let R and R* be sequential de priori

information such that R=(W1 at prevk)&(W

2 at prevk+1)&... &(W

n at prevk+n-1) and

R*=(W*1 at prevk)&(W*

2 at prevk+1)&... &(W*

n at prevk+n-1). In a good sense, R ascribes

w1 to the k epistemic moments earlier time, w

2 to the k+1 epistemic moments earlier

time, …, wn+1

to the k+n-1 epistemic moments earlier time; similarly for R*. Then,

102102102102

(11) R* is better de priori information than R iff for any k∈1,...,m,

w*k⊆w

k, and for some k∈1,...,m, w*

k⊂wk.

In other words, R* is better than R exactly when R* attributes at least equally narrow

intervals to the mentioned epistemic moments and R* attributes a strictly narrower

interval to one of the epistemic moments than R. Let R be any sequential de priori

information. Then, for any probability function C and tensed propositions X and Y,

(12) R is well-specified de priori information with respect to <C, X, Y>

iff C(X/Y&R)=C(X/Y&R*) for any sequential de priori information R*

better than R.

When this condition is satisfied, I will often say informally that C(X/Y&R) is conditioned

upon well-specified de priori information. This definition allows us to formulate the

desired rules without precisely specifying what times it had been until the moment from

which updating occurs.

Now, I am ready to formulate my first updating rule in this chapter, “General

Shifted Strict Conditionalization”: Consider a sequence of observations E1, E

2, ...E

m, a

sequence of intervals v1, v

2, ... v

m, and another sequence of intervals w

1, w

2, ... w

n+1.

Assume that (a) the truth-value of X is invariant within vm

and that of Ek is invariant

within vk for each k∈1,...,m and (b) Cn(X in v

m/(E

1 in v

1)&... &(E

m in v

m)&(W

1 at

prevm)&...& (Wn+1

at prevm+n)) is conditioned upon a well-specified temporal description.


103103103103

(GSSC) Cn+m(X)=Cn(X in vm

/(E1 in v

1)&...&(E

m in v

m)&(W

1 at prevm)&...& (W

n+1

at prevm+n)) if

(i) B is sure at tn+m that for each k∈1,...,m, [Ek was/is true and it was/is

vk] m-k epistemic moments ago, and

(ii) B is sure at tn+m that for any k∈1,...,n+1, [it was wk] m+k-1 epistemic

moments ago,

where Cn and Cn+m are B’s credence functions at tn and tn+m. Less formally, we can

rewrite SSSC in this way: Let E be (E1 at prevm-1)&...&(E

m at prev0), V be (V

1 at prevm-1)

&...&(Vm

at prev0), and W be (W1 at prevm)&(W

2 at prevm+1) &…&(Wn+1

at prevm+n).

Assume that (a) and (b) are true. Then,

(GSSC) Cn+m(X)=Cn(the de-indexicalization of X under Vm

/the sequential de-

indexicalization of E under V & the sequential re-indexicalization of W for

the m epistemic moments earlier time) if B has certainly learned until tn+m

that E&V&W is true.

Unfortunately, GSSC is a principle with an extremely narrow range of

application. In order to use it, we need to know what time it has been since our very first

observation. Hence, we are forced to move to our next, more general updating principle.

First, let me outline the core idea: Let Eo&Vo&Woo∈O be a partition such that Eo=(E1

o at

prevm-1)&...& (Em

o at prev0), Vo=(V1

o at prevm-1)&...&(Vm

o at prev0), and Wo=(W1

o at

104104104104

prevm)&...& (Wn+1

o at prev0), for each o∈O. I consider each Eo&Vo to represent a possible

evidential scenario and each Wo to represent what times it had been until tn. For

simplicity, let O be 1, 2, ...p. By GSSC:

Cn+m(X) would be

Cn(X in vm

1/D1&R1) if B were sure at tn+m of E1&V1&W1,

Cn(X in vm

2/D2&R2) if B were sure at tn+m of E2&V2&W2,

...

Cn(X in vm

p/Dp&Rp) if B were sure at tn+m of Ep&Vp&Wp,

where Do is the sequential de-indexicalization of Eo under Vo and Ro is the sequential re-

indexicalization of Wo for the m epistemic moments earlier time for each o∈O. If we

accept GSSC (at least for its narrow range of application), it is natural that B’s credence

at tn+m in X is the weighted average of values on the right-hand sides of the above

equations with the weights coming from B’s credences at tn+m in Eo&Vo&Wo.

To implement this idea, we need to finish some formal homework first: Consider

a partition &1≤k≤m((Eko&V

ko) at prevm-k)&&1≤k≤n+1(W

ko at prevm+k-1)o∈O such that (i)

Cn+m(&1≤k≤m((Eko&V


ko at prevm+k-1))>0 for each o∈O and (ii)

∑o∈OCn+m(&1≤k≤m((Eko&V


ko at prevm+k-1))=1 where Cn+m is an

agent B’s credence function at tn+m. (The intended interpretation of this partition is that

each member is a hypothesis about (i) what observations have been made at what times

after tn and (ii) what times it had been until tn.) I will call any member of this partition

105105105105

“(B’s) general time-observation proposition from tn to tn+m.” If that partition also satisfies

the condition that for each o∈O, Cn(&1≤k≤m(Eko in v

ko)&&1≤k≤n+1(W

ko at prevk-1))>0,

then I will call the partition “(B’s) general time-observation partition from tn to tn+m.”

Now, I am ready to present my next updating rule, called “General Shifted

Jeffrey Conditionalization”: Let &1≤k≤m((Eko&V

ko) at prevm-k) & &1≤k≤n+1(W

ko at

prevm+k-1)o∈O be B’s sequential time-observation partition from tn to tn+m over [t0,tn+m].

Assume that for any o∈O (a) the truth-value of X is invariant within vm

o and that of Eko is

invariant within each vko for any k∈1,...,m and (b) Cn+m(&1≤k≤m((E

ko&V

ko) at prevm-

k)&&1≤k≤n+1(Wko at prevm+k-1)) is conditioned upon a well-specified temporal description.

Then,

(GSJC) Cn+m(X)=Σo∈O[Cn(X in vm


ko)&&1≤k≤n+1(W

ko at prevk-1))

Cn+m(&1≤k≤m((Eko&V


ko at prevm+k-1))],

where Cn and Cn+m are B’s credence functions at tn and tn+m. Less formally: Let

Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and Wo=&1≤k≤n+1(W

ko at prevm+k-1)

for each o∈O, so that Eo&Vo&Woo∈O is the same as &1≤k≤m((Eko&V

ko) at prevm-

k)&&1≤k≤n+1(Wko at prevm+k-1)o∈O, B’s general time-observation partition from tn to tn+m.

Assume that for each o∈O (a) and (b) are true. Then,

106106106106

(GSJC) Cn+m(X) = the weighted average of Cn(the de-indexicalization of X under

Vm

o/the sequential de-indexicalization of Eo under Vo & the sequential re-

indexicalization of Wo) with the weights coming from Cn+m(Eo&Vo&Wo)

where Cn and Cn+m are B’s credence functions at tn and tn+m. I think this is a natural

generalization of GSSC for when the agent is not sure of what sequence of evidence she

has received at what times and what times it had been before receiving any sequence of

evidence in consideration.

To see how GSSC and GSJC work, consider the following example: Example 3.

Let R be the tensed proposition expressed by “it rains today in Boston,” and let P be that

expressed by “there is a form of precipitation today in Boston.” Jane is born on Saturday,

knowing that (SAT) it is Saturday.

Just after her birth, she falls asleep and then wakes up on Sunday. She learns that

(SUN) it is Sunday. At this time, Jane assigns the credence of 0.8 to [R on Monday] given

[P on Monday]. Immediately after waking up, she takes a sleeping pill that will make her

wake up either on Monday or Tuesday, but she will not know which day it is when she

wakes up.

In fact, Jane wakes up on Monday. On being awakened, she is told that there is a

form of precipitation today. For the mentioned reason, she does not know whether (MON)

it is Monday or (TUE) it is Tuesday; indeed, she assigns the credence of 0.5 to each of

MON and TUE. (Hence, 0.5 is her credence at this moment in P&MON.) Later on that

day, she is told that it is Monday.

107107107107

For brevity, let b be the moment of Jane’s birth on Saturday, let s be the moment

of waking up on Sunday, let m be the moment of waking up on Monday, and let m+ be

the moment of being told that it is Monday. We assume that she does not observe

anything between these moments. What, then, is her rational credence at m+ in R?

First, let’s focus on her credal transition from m to m+. By GSSC,

(13) Cm+(R)=Cm(R on Monday/(MON on Monday)&(MON at

prev0)&(SUN at prev1)&(SAT at prev2)).

The conditional part of the right-hand side was acquired as in Figure 10:

We can simplify (13) by using these facts: Jane was sure at m that MON is true on

Monday, it was Sunday at the last epistemic moment (=s), and it was Saturday two

epistemic moments ago (=b). At m, she believed MON to the degree of 1/2. Hence,

(14) Cm+(R)=Cm(R on Monday/MON)=2Cm((R on Monday)&MON).

Time b s m m+

An observation at m MON

Temp. desc. until m SAT at prev2 SUN at prev1 MON at prev0

SAT at prev2 SUN at prev1 MON at prev0MON on Monday

Sequential re-indexicalization

de-

indexicali

zations

Figure 10: Reindexicalization and Deindexicalization 1. The sequential de

priori information (MON at prev1)&(SUN at prev2)&(SAT at prev3) is re-

indexicalized to (MON at prev0)&(SUN at prev1)&(SAT at prev2) and the

108108108108

Let R*=(R on Monday)&MON; thus, Cm+(R)=2Cm(R*). To use this equation, we need to

find her credence at m in R*.

Focus on Jane’s credal transition from s to m: She is not sure at m whether it is

Monday or Tuesday, but she remembers that it was Sunday at the previous moment and

that it was Saturday two moments ago. By GSJC,

(15) Cm(R*)=

Cs(R* on Monday/(P on Monday)&(SUN at prev0)&(SAT at prev1))

Cm((P&MON)&(SUN at prev1)&(SAT at prev2) )+

Cs(R* on Tuesday/(P on Tuesday)&(SUN at prev0)&(SAT at prev1))

Cm((P&TUE)&(SUN at prev1)&(SAT at prev2)).

For [P on Monday] is the de-indexicalization of P under MON, and [P on Tuesday] is

that of P under MON, and (SUN at prev0)&(SAT at prev1) is the re-indexicalization of

(SUN at prev1)&(SAT at prev2) for the previous epistemic moment, as in Figure 11. Also,

observe these facts: (i) [R* on Monday] is equivalent to [R on Monday].45

(ii) [R* on

Tuesday] is impossible.46

(iii) Jane was sure at s of [SUN at prev0]&[SAT at prev1]. (iv)

Jane is sure at m of [SUN at prev1]&[SAT at prev2]. Hence,

45 For [((R on Monday)&MON) on Monday]=[((R on Monday) on Monday)&(MON on Monday)]=(R on

Monday).

46

For [((R on Monday)&MON) on Tuesday] entails (MON on Tuesday), which is a contradiction.

109109109109

(16) Cm(R*)=Cs(R on Monday/P on Monday)Cm(P&MON)=0.4.

From (14) and (16), it follows that her credence at m+ in R is 0.8. This is an intuitive

result, as her credence at s in raining on Monday was 0.8, conditional on precipitation on

Monday. She later learns that there is a form of precipitation today and that today is

Monday. Hence, it follows that there is a form of precipitation on Monday after all. Given

these facts, it is natural that her credence at m+ (∈ Monday) is 0.8 in R.

Finally, I want to formulate GSJC and GSSC in more succinct forms: Let

Eo&Vo&Woo∈O be an agent B’s general time-observation partition from tn to tn+m,

where Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and Wo=&1≤k≤n+1( W

m+ko at

Time b s m

An observation at m P

Temp. desc. until m SAT at prev2 SUN at prev1

The 1st seq. re-indexicalization

& de-indexicalizationSAT at prev1 SUN at prev0 P on Monday

The 2nd seq. re-indexicalization

& de-indexicalizationSAT at prev1 SUN at prev0 P on Tuesday

Seq. re-indexicalization

de-

indexicali

zations

Figure 11: Reindexicalization and Deindexicalization 2. The observation P is de-

indexicalized to (P on Monday) and (P on Tuesday), and sequential de priori

information (SUN at prev1)&(SAT at prev2) is re-indexicalized to (SUN at

prev0)&(SAT at prev1).

110110110110

prevm+k-1). Given this partition, I introduce these definitions: (i) Eo&Vo&Woo∈O is

probabilistically optimal for X iff for any o∈O, Cn+m(X in vm

o/Do&Ro) is conditioned

upon a well-specified temporal description, where Do=&1≤k≤m(Eko in v

ko) and

Ro=&1≤k≤n+1(Wko at prevk-1). Also, remember the definition of logical optimality in the

last chapter. Given these two definitions, (ii) Eo&Vo&Woo∈O is optimal for X iff

Eo&Voo∈O is logically optimal for X and Eo&Vo&Woo∈O is probabilistically optimal

for X. Then,

(GSJC) Cn+m(X)=Σo∈OCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo) if Eo&Vo&Woo∈O

is optimal for X,

where Do=&1≤k≤m(Eko in v

ko) and Ro=&1≤k≤n+1(W

ko at prevk-1). Next, let E&V&W be

B’s sequential time-observation partition from tn to tn+m, where E=&1≤k≤m(Ek at prevm-k),

V=&1≤k≤m(Vk at prevm-k) and W=&1≤k≤n+1(W

m+k at prevm+k-1). Then,

(GSSC) Cn+m(X)=Cn(X in vm/D&R) if E&V&W is optimal for X,

where D=&1≤k≤m(Ek in v

k) and R=&1≤k≤n+1(W

k at prevk-1). Obviously, these

reformulations are equivalent to the original.

111111111111

So far, I have presented GSJC and GSSC, illustrated how they work with an

example, and provided shorter formulations for them. The next step is to defend the new

updating principles with a new variant of the Conditional Expert Principle.

D. Temporal Conditional Multiple Expert Principle

In the last chapter, I introduced the Temporal Conditional Expert Principle. That principle

described an epistemic relation between two agents located at different times. As such, it

takes only their times and observations into consideration. In this section, I will present a

new principle, which is similar to TCE but assumes additional agents contributing to the

epistemic cooperation.

First, consider two agents Bn+m, located at time tn+m, and Bn, located at time tn,

where tn+m>tn; for convenience, I will call Bn+m “the client” and Bn “the expert.” As the

names indicate, the client wants to set her credences at tn+m by consulting Bn’s credal

opinion at tn. (I will often omit “at tn+m” and “at tn.”) Here, I suppose that the expert’s

credal judgment is not dependent upon any other agent’s data or judgment. See Figure 12.

Call this type of situation “a two agent situation.” In this situation, what is the rational

way for the client to set her credences by checking the expert’s?

Figure 12: Judgmental Dependence. The arrow line indicates the judgmental

dependence. There is no informational dependence on any other agents.

112112112112

Earlier, I argued for the following answer: For any tensed proposition X,

(TCE) Cn+m(X/E&V &prn(X′/E′)=r)=r if the left-hand side has a defined value

and the following conditions are satisfied:

a) Cn+m is the client’s credence function at tn+m, and prn is the expert’s

credence function at tn.

b) All information had at tn+m by the client is accessible to the expert

at tn, possibly except E, and the expert perhaps does not know at tn

whether E′ is true.

c) X′ and E′ are tensed propositions such that the client knows at tn+m

that if V is true, [X is presently true iff X′ was true at tn] and [E is

presently true iff E′ was true at tn].

I briefly repeat the rationale that I provided for this principle: By (a), it seems rational

that Cn+m restricts Cn given the agent’s intention to consult the expert’s opinion in order

to set her de nunc credences. By (b), it will however be irrational for the client to set her

credence in X to be simply the same as the credence that the expert assigned to X. By (c),

the client will think that if V is true, it is best to assign r to X conditional on E provided

that Ex’s credence in X′ given E′ is r; for, she knows that if V is true, [X is presently true

iff X′ was true when the expert had the consulted credal opinion] and [E is presently true

iff E′ was true when the expert had the consulted credal opinion].

I still believe that TCE makes sense when there are no agents other than the

client and the expert to take into consideration. However, think about the following

113113113113

situation: Consider n+m+1 agents, Bn+m, …, Bn, …, B0, located at different moments,

tn+m, …, tn, …, t0, where tn+m>…>t0; for convenience, we will call Bn+m “the client,”

Bn+m, …, Bn+1 “the direct data providers,” Bn “the expert,” and Bn, …, B0 “the indirect

data providers.” We suppose that the client intends to judge the probability of X with the

help of the expert, given the data observed by the direct data providers. Also, the expert’s

judgment, consulted by the client, is based upon the data observed by the indirect data

providers. See Figure 13. Call this situation “a multiple agent situation.”47

In this

situation, what is the rational way for the client to set her present credence?

As an answer, I suggest a principle that I call “the Temporal Conditional

Multiple Expert Principle”: Let E be a tensed proposition specifying the observations that

47 In their paper (forthcoming), Dietrich and List discuss the topic of how one can aggregate the opinions

of multiple agents in a rational way. I believe that such a theory will be useful also for clarifying how a

single person can aggregate the opinions of her multiple selves in the past in a rational way. Another

important topic, which has been left unstudied as far as I know, is that of how an individual is supposed to

aggregate the de se opinions of multiple agents.

Figure 13: Direct and Indirect Data Providers. Here, Bn+m is the client and Bn is

the expert. The dashed arrow line indicates the informational dependence, while

the solid arrow line—from Bn+m to Bn—indicates the judgmental dependence.

114114114114

have been made by the direct data providers, let V be a tensed proposition about in what

times the direct data providers are located, and let W be a tensed proposition about in

what times the indirect data providers are located. Then, for any tensed proposition X,

(TCME) Cn+m(X/E&V&W&prn(X′/E′&W′)=r)=r, if the left-hand side has a defined

value and the following conditions are satisfied:

(d) Cn+m is the client’s credence function at tn+m and prn is the expert’s

credence function at tn.

(e) All information had at tn+m by the client is accessible to the expert

at tn possibly except E, and so the expert perhaps does not know at

tn whether E′ is true.

(f) X′ and E′ are tensed propositions such that the client knows at tn+m

that if V is true, [X is presently true iff X′ was true at tn] and [E is

presently true iff E′ was true at tn].

(g) W′ is a tensed proposition such that B knows at tn+m that [W is

presently true iff W′ was true at tn].

Here, the main difference is (d): According to TCE, the client does not have to take the

temporal locations of the agents providing data to the expert. According to TCME, the

client needs to take those agents’ temporal locations into consideration; conditional on

the assumption that the indirect data providers were located at the times specified by W,

the client needs to consult the expert’s credence in X′ not only conditioned upon E′, but

115115115115

also upon W′, where W′ specifies the same temporal locations of the indirect data

providers as specified by W.

Why this difference? In a multiple agent situation, the client will be aware that

the expert’s credal opinion was made by depending upon the data from the indirect data

providers. In order for the expert to correctly interpret those data, he will need the

information about when those data were observed; in other words, he will need the

information about the indirect data providers’ temporal locations.

To appreciate this point, let’s consider an analogous example: Example 4. Four

meteorologists are flying in balloons in the New England sky. Let’s call them “B3,”

“B2,”“B1,” and “B0,” in the order of spatial proximity to B3. Suppose that B3 is judging the

probability of rain in her region with the help of the other meteorologists. I assume that

each Bk is equipped with a walkie-talkie but does not communicate with the other

meteorologists unless Bk is contacted by a Bi>k or needs the data or judgment of some Bi<k.

(Hence, the information flows from B0 to B3 but not in the other direction.) In this

situation, what will be the best strategy for B3 to make a credal judgment about rain in her

region?

One good strategy would be for her to “delegate” some required tasks to, say, B1,

so that while B3 gathers data from B3 and B2, she depends upon the judgment of B1. (Note

that this is not to disregard the observations made by B1 and B0 because, if rational, B1

will judge by taking their observations into consideration.) Let’s call B3 “the client,” B3

and B2 “the direct data providers,” B1 “the expert,” and B1 and B0 “the indirect data

providers.”

116116116116

First, the client will need the information about the direct data providers’ spatial

locations in order to correctly judge the probability of rain using their data. This is

because without knowing where the observed events are occurring, it will be difficult to

correctly judge the relevance of the observations made by the direct data providers to the

possibility of rain in her region.48

For example, suppose that E represents the data

observed by the direct data providers, where E=(5°C temperature is being observed by

B3)&(thick clouds are being observed by B2). To correctly interpret E, the client (=B3)

will need information about where the direct data providers observed the conjuncts of E,

such as L=(B3 is located in Amherst)&(B2 is located in Pelham).

Second, the client also will need the spatial locations of the indirect data

providers in order to rationally utilize the expert’s credal judgment. Intuitively, given

E&L, the client will consult the expert’s probability of rain in Amherst conditioned upon

E ′, where E ′=(5°C temperature is being observed in Amherst)& (thick clouds are being

observed in Pelham). Now, if the expert is rational, he will make this credal judgment on

the basis of the data from the indirect data providers. Consequently, the expert will need

the spatial locations of the indirect data providers to correctly interpret the data from

them, just as the client needs the spatial locations of the direct data providers to correctly

interpret their data.

To see this point clearly, suppose that the expert has the information that (F) a

strong wind is observed by B1, and the wind is observed to be blowing from east to west

by B0. Then, compare two possible spatial locations of B1 and B0:

48 Also, she will need to know the region in which she is located to know the region for which she is

judging the probability of rain.

117117117117

(M) B1 is located in Belchertown, and B0 is located in Ware.

(M*) B1 is located in Newton, and B0 is located in Cambridge.

From the point of view in Amherst, a strong east wind in nearby eastern areas such as

Belchertown and Ware will elevate the probability of rain in Amherst given E′, but the

same wind in distant lands like Newton and Cambridge will not be particularly relevant

to the possibility of rain in Amherst on the condition of E′.49 Hence,

(17) prn(raining in Amherst/E ′&M)≠prn(raining in Amherst/E ′&M*),

where prn is the expert’s credence function. Now, assume that the expert was almost sure

of M*, so that

(18) prn(raining in Amherst/E′)≈prn(raining in Amherst/E′&M*),

but also that the client is quite sure of M. In that case, it will be irrational that

(19) Cn+m(raining /E&L &prn(raining in Amherst/E′)=r)=r if defined,

49 For the relevant geographical data, see the entry of “Massachusetts” in http://www.wikipedia.org.

118118118118

where Cn+m is the client’s credence function. For the expert’s above conditional credence

was, from the client’s point of view, a judgment largely based upon wrong information

about the indirect data providers’ spatial locations.

This problem is due to the potential difference between the client’s and the

expert’s opinions about the indirect data providers’ spatial locations. One way to bracket

out this difference is to consult the expert’s credence conditioned upon the (from the

client’s point of view) correct information about the spatial locations of the indirect data

providers. This suggests the following relation between the client’s and the expert’s

credal opinions: For any real number r,

(20) Cn+m(raining/E&L&M&prn(raining in Amherst/E′&M)=r)=r if

defined.

It is not difficult to apply the same idea to a case in which the agents are located at

different times; consider example 5. Four meteorologists have made observations about

the weather of Amherst on different days. Call the meteorologists “B3,” “B2,”“B1,” and

“B0,” in the reverse order of time. We suppose that B3, “the client,” is making a credal

judgment about whether it will rain today with the help of the other three agents. In

particular, she depends upon the data from B3, B2, “the direct data providers,” and the

judgment of B1, “the expert.” It is clear that she also comes to indirectly rely upon the

data from B1, B0, “the indirect data providers.” In this case, what will be the correct way

for the client to make her credal judgment about whether it will rain in Amherst today?

119119119119

First, she will need to know the temporal locations of the direct data providers in

order to correctly interpret the data from them. Let E=(5°C temperature is being observed

by B3)&(thick clouds were observed by B2), and V=(B3 is located in Saturday)&(B2 is

located in Sunday). Given V, the client can process E into the equivalent data E′=(the 5°C

temperature is being observed on Sunday)&(thick clouds were observed on Sunday).

Second, she will need to know the temporal locations of the indirect data

providers in order to correctly utilize the expert’s credal judgment. For the expert’s credal

opinion must have been produced on the bases of their data. Suppose that the expert had

the information that (F) B0 had met somebody telling him that for the next three days,

once thick clouds have formed, they will not go away quickly, and B1 was told that the

guy she met yesterday was a very good expert about cloud forming. Consider these

temporal locations of the indirect data providers:

(W) B1 was located on Saturday, and B0 had been located on Friday.

(W*) B1 was located on Thursday, and B0 had been located on Wednesday.

Clearly,

(21) prn(raining on Monday/E′&W)≠prn(raining on Monday/E′&W*),

where prn is the expert’s credence function. Assuming that the client is almost sure of W,

but the expert is almost sure of W*, it will be irrational that

120120120120

(22) Cn+m(raining /E&V&prn(raining on Monday/E′)=r)=r if defined,

where Cn+m is the client’s credence function. For the expert’s above conditional credence

is largely based on, from the client’s point of view, wrong information about the indirect

data providers’ temporal locations, which was essential to judging correctly E′’s

relevance to whether or not it will rain on Monday. Rather, the rational way that the client

would utilize the expert’s opinion is the following:

(23) Cn+m(raining /E&V&W&prn(raining on Monday/E′&W)=r)=r if

defined.

Until now, we have discussed cases in which the client has a means of specifying

the indirect data providers’ locations in non-indexical ways. However, I do not think the

lesson we have learned from these examples depends upon the existence of such a non-

indexical method of specifying the indirect data providers’ locations. So let W and W′ be

the tensed propositions specifying the indirect data providers’ temporal locations such

that from the client’s point of view, (*) W is presently true iff W′ was true at the expert’s

time. Then, it will be the case that

(24) Cn+m(X/E&V&W&prn(X′/E′&W′)=r)=r if defined,

where X′ and E′ satisfy TCME’s provisos (a)-(c) with respect to X, E, and V. For W tells

the client that W′ was [the tensed proposition specifying the indirect data providers’

121121121121

temporal locations] that was true at the expert’s time. By introducing (*) as the fourth

proviso (d), we acquire the general principle TCME.

In this section, I have argued that we need to modify TCE in order to capture the

rational method for an agent to defer to the credal judgment of another agent located at an

earlier moment, where both agents are provided relevant data.

E. A Defense of GSJC

In this section, I will present a new principle, “General Shifted Sequential Rigidity”

(hereafter: GSR), and defend it by making use of TCME. After defending it, I will point

out that GSR entails GSJC. Since GSSC is a special case of GSJC, the two principles

presented in the last section will have been defended.

I start by presenting GSR: Let Eo&Vo&Woo∈O be B’s general time-observation

partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and

Wo=&1≤k≤n+1(Wko at prevm+k-1). Next, suppose that Eo&Vo&Woo∈O is optimal for X.

Then,

(GSR) Cn+m(X/Eo&Vo&Wo)=Cn(X in vm

o/Do&Ro),


indexicalization of Wo for the m epistemic moments earlier time. (In other words,

122122122122

Do=&1≤k≤m(Eko in v


ko at prevk-1).) It is not difficult to see that

GSR entails GSJC.50

How do we defend GSR? We defend it by using TCME. First, let Bn+m, …, B0 be

the agent B’s selves at tn+m, …, t0, where tn+m>…>t0. We suppose that Bn+m sets her

credence in X with the help of Bn+m-1, …, B0. One way to do so will be to collect the data

from Bn+m, …, Bn+1 and delegate the job of making a suitable conditional credal judgment

to her past self, say, Bn. Accordingly, let’s call Bn+m “the client” and Bn “the expert.”

One important question is whether this is a two agent situation or a multiple

agent situation. Clearly, this is a case of the latter. Not only does the client depend upon

Bn+m, …, Bn+1 to acquire extra data, but the expert must also depend upon Bn, …, B0 to

make the suitable conditional credal judgment. Accordingly, let’s call Bn+m, …, Bn+1 “the

direct data providers” and Bn, …, B0 “the indirect data providers.”

Once put in this way, it is plausible that TCME applies to this case, since I have

previously argued that in a multiple agent case, TCME, not TCE, is the principle

describing the epistemic relation between the client and the expert. Then, we can derive

the principle that I call the “General Shifted Backward Reflection Principle” from TCME:

Let X be any tensed proposition and Eo&Vo&Woo∈O be B’s general time-observation

partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and

Wo=&1≤k≤n+1(Wko at prevm+k-1). Next, let Dο be the sequential de-indexicalization of Eo

50 Suppose GSR. So Cn+m(X/Eo&Vo&Wo)=Cn(X in v

mo/Do&Ro) for any o∈O. Since Eo&Vo&Woo∈O is a

general time-observation partition from tn to tn+m, (i) ∑o∈OCn+m(X/Eo&Vo&Wo)=1 and (ii) (X/Eo&Vo&Wo)>0

for each o∈O. Thus, Cn+m(X)=∑o∈OCn+m(X&Eo&Vo&Wo)=∑o∈OCn+m(X/Eo&Vo&Wo)Cn+m(Eo&Vo&Wo)= (by

supposition) ∑o∈OCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo). Done.

123123123123

under Vo (i.e., Dο=&1≤k≤m(Eko in v

ko)), and let Ro be the sequential re-indexicalization of

Wο for the m epistemic moments earlier time (i.e., Wo= &1≤k≤n+1(Wko at prevk-1)).

Assume that Eo&Vo&Woo∈O is optimal for X. Then,

(GSBR) Cn+m(X/Eο&Vο&Wο&Cn(X in vm

o/Dο&Rο)=r)=r if defined.

For (a) Cn+m is the client’s credence function, and Cn is the expert’s, (b) Eo specifies the

data accessible to the client but perhaps not to the expert, (c) the client knows (at tn+m)

that X is presently true iff [X in vm

o] was true m epistemic moments ago, and Eo is

presently true iff Do was true m epistemic moments ago, and (d) the client also knows (at

tn+m) that Wo is presently true iff Ro was true m epistemic moments ago. Since the

elements composing GSBR satisfy the provisos of TCME, GSBR is a special case of

TCME. GSR is derivable from GSBR under the presupposition that B always remembers

her past credence function with perfect confidence and correctness.51

Remember that GSBR entails GSR and GSR entails GSJC. Therefore, we have

good reason to accept GSJC.

F. Too Far Past Does Not Matter

The earlier formulation of GSJC has a practical problem: In most credal transitions, we

do not worry about what time it was at t if t is a moment sufficiently far past. Even in

such a case, GSJC asks us to take such a matter into consideration. In this section, I will

51 Let r=Cn(X in vj/Do&Ro). By the presupposition of perfect memory, Cn+m(Cn(X in vj/Do&Ro))=r)=1.

Thus, Cn+m(X/Eo&Vo&Wo)=Cn+m(X/Eo&Vo&Wo&Cn(X in vj/Do&Ro)=r)=(by GSBR)r=Cn(X in vj/Do&Ro).

Done.

124124124124

provide another formulation of GSJC, which is (i) relatively free from this problem and

(ii) still equivalent to the original formulation of GSJC.

First, I review GSJC: Let Eo&Vo&Woo∈O be an agent B’s general time-

observation partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at

prevm-k), and Wo=&1≤k≤n+1(Vko at prevm+k-1). Then,

(GSJC) Cn+m(X)=Σo∈OCn(X in vm


is optimal for X,

where Do is the sequential de-indexicalization of Eo under Vo (i.e., Do=&1≤k≤m(Eko in v

ko))

and Ro is the sequential re-indexicalization of Wo for the m epistemic moments earlier

time (i.e., Ro=&1≤k≤n+1(Wko at prevk-1)).

We can expect a complaint: If I update in accordance with GSJC, then I have to

presently assign to X the weighted average of [X in vm

o] given Do&Ro (with the weights

coming from…), where Ro specifies what times it had been at all my epistemic moments

until that from which I am updating. These epistemic moments even include the moment

of my first observation. This appears to be absurdly demanding. For example, why must I

take my birth time into consideration, in judging the probability of rain today?

This is a fair complaint. There should be a way in which one can rationally judge

the probability of rain today without worrying about when she was born, when she fell in

love for the first time, etc. So, I reformulate GSJC into a principle that does not mention

125125125125

epistemic moments that are too far in the past to be relevant. First, I suggest a modified

definition of general time-observation partition. The core idea is that first we can remove

from the given general time-observation partition its elements ascribing times to

epistemic moments too far in the past to be relevant, and second, we can formulate the

updating principle that works with the remaining partition.

Here is the first step: Again, let Eo&Vo&Woo∈O be an agent B’s general time-

observation partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at

prevm-k), and Wo=&1≤k≤n+1(Wko at prevm+k-1). Consider some i∈1,...,n+1. Given

Eo&Vo&Woo∈O and i, let Ep&Vp&Wpp∈P be the maximal partition such that for each

p∈P, there exists o∈O such that Ep=Eo, Vp=Vo, and Wp=&1≤k≤n-i(Wko at prevm+k-1).

52

Given these partitions, I will say that Ep&Vp&Wp is the abbreviation of Eo&Vo&Wo. It

follows that for any o∈O, there exists p∈P such that Eo&Vo&Wo=(Ep&Vp&Wp&W*p),

where W*p=&n-i +1≤k≤n+1(Wm+k

o at prevm+k-1). In such a case, I will say that W*p is a

complement of Ep&Vp&Wp for Eo&Vo&Wo. The point is that Ep&Vp&Wpp∈P is the

same as Eo&Vo&Woo∈O except that each Wp may include only a fragment of

corresponding Wo. In such a case, I will say that Ep&Vp&Wpp∈P is (B’s) general time-

observation partition from tn to tn+m over [ti,tn+m].53

Also, I will say that Ep&Vp&Wpp∈P

52 Here, I suppose that if “1≤k≤n-i” is not satisfied by any number k, then &1≤k≤n-i(W

ko at prevm+k-1) is

vacuously true. For example, if i=n+1, &1≤k≤n-i(Wko at prevm+k-1)=&1≤k≤-1(W

ko at prevm+k-1)=T, where T is any

tautology.

53

Note: If i=n+1, then Wp is vacuous (see the previous footnote) and so Ep&Vp&Wpp∈P=Ep&Vpp∈P=

Eo&Voo∈O.

126126126126

is sufficiently inclusive for a tensed proposition X iff for each o∈O and p∈P, if

Ep&Vp&Wp is an abbreviation of Eo&Vo&Wo and W*p is the complement, then Cn(X in

vm

p/Dp&Rp)= Cn(X in vm

p/Dp&Rp&R*p), where Dp is the sequential de-indexicalization

of Ep under Vp and Rp and R*p are the sequential re-indexicalizations of Wp and W*p for

the m epistemic moments earlier time.

Here is the second step: Let Ep&Vp&Wpp∈P be a general time-observation

partition from tn to tn+m over [ti,tn+m]. Then,

(GSJC-) Cn+m(X)=Σp∈PCn(X in v

mp/Dp&Rp)Cn+m(Ep&Vp&Wp) if Ep&Vp&Wpp∈P is

optimal and sufficiently inclusive for X,

where Dp is the sequential de-indexicalization of Ep under Vp, and Rp is the sequential re-

indexicalization of Wp for the m epistemic moments earlier time. Of course, it is possible

to formulate the corresponding variant of GSSC: Let E&V&W be a general time-

observation partition from tn to tn+m over [ti,tn+m]. Then,

(GSSC-) Cn+m(X)=Cn(X in v

mp/D&R) if E&V&W is optimal and sufficiently

inclusive for X,

where D is the sequential de-indexicalization of E under V, and R is the sequential re-

indexicalization of W for the m epistemic moments earlier time. It is provable that GSJC-

is equivalent to GSJC and that GSSC- is equivalent to GSSC. (See APPENDIX A.)

127127127127

In GSJC- and GSSC

-, epistemic moments that are too far past are not mentioned

as long as “What times were it in those moments?” is an irrelevant question to how

probable the target tensed proposition is now. Hence, we now have more practical

variants of GSJC and GSSC.

G. Application to the SB Problem

So far, I have developed a more general rule for de nunc updating than SSJC. This

generalization was initially motivated by the fact that SSJC does not apply correctly to

SB’s credal transition from m to m+. Hence, it will be interesting to see how well our new

rule, GSJC, will do with respect to the same credal transition.

For an easier discussion, I will first apply GSJC to a variant of the SB problem.

The lessons from the variant problem will help us to understand how to apply GSJC to

the original problem. Think about this version of the SB problem: SB problem 3. SB is

born on Sunday, knowing that it is Sunday. She also knows the following events will

happen during the next three days: Immediately after her birth, a group of evil

experimenters put her to sleep. Next, they toss a fair coin. Case 1: (H) The coin lands

heads. Then, they wake her up once on Monday. Case 2: (T) The coin lands tails. In this

case, the experimenters wake her up twice, the first time on Monday and the second time

on Tuesday. Between the two awakenings, they inject her with a drug that erases her

memory of the first awakening. In either case, SB is told that it is Monday one minute

after she wakes up on Monday. Here is the question: What is her credence in H when she

is told that it is Monday?

Let s be the moment of her birth on Sunday, m be the moment of her wakeup on

Monday, and m+ be the moment of being told that it is Monday. There are two ways in

128128128128

which we can answer the above question by using GSJC. First, we can use GSSC for her

credal transition from m to m+:

(25) Cm+(H)=Cm(H on Monday/(MON on Monday)&(MON at

prev0)&(SUN at prev1)).

Since H is a genuine proposition, [H on Monday] is equivalent to H. It is a tautology that

MON is true on Monday. Finally, when she wakes up on Monday, she remembers that it

was previously Sunday. Hence,

(26) Cm+(H)=Cm(H /MON at prev0)=Cm(H/MON)=

Cm(H&MON)/Cm(MON).

To find the last value, we use GSJC for her credal transition from s to m. At m, SB is sure

that she is observing W at that moment and that it was previously Sunday, but she does

not know whether it is Monday or Tuesday. Thus, her general time-observation partition

from s to m is W&(MON at prev0)&(SUN at prev1), W&(TUE at prev0)&(SUN at prev1).

Hence, the correct instance of GSJC is

(27) Cm(H&MON)=

Cs((H&MON) on Monday/(W on Monday)&(SUN at prev0))*

Cm(W&(MON at prev0)&(SUN at prev1))+

129129129129

Cs((H&MON) on Tuesday/(W on Tuesday)&(SUN at prev0))*

Cm(W&(TUE at prev0)&(SUN at prev1)).

Now observe the following facts: First, [(H&MON) on Monday] is clearly equivalent to

H. Second, on Sunday, SB knew that it was Sunday, and she fully expected to wake up

on Monday. Third, H&MON cannot be true on Tuesday. Thus, we can simplify (27) into

(28) Cm(H&MON)=Cs(H)*Cm(W&(MON at prev0)&(SUN at prev1)).

On Sunday, SB believed to the degree of ½ that the coin would land heads. On Monday,

she certainly knows that she is waking up with such and such a memory, and she

remembers that it was previously Sunday. Therefore,

(29) Cm(H&MON)=1/2Cm(MON at prev0)=1/2Cm(MON).

It follows from (26) and (29) that her credence at m+ in H is ½. (Note that this result

captures the intuition that I described in Section C.)

Second, GSSC provides the following instance for SB’s credal transition from s

to m+: When SB is told that it is Monday, she remembers that she previously experienced

waking up with the memory of Sunday as the last memory, and she learns that it is

Monday now. Hence, she is sure at m+ of (MON at prev0)&(W at prev1). Also, she is sure

at that moment that it is Monday then and that it was Monday previously. Thus, she is

130130130130

sure at m+ of (MON at prev0)&(MON at prev1). Finally, she remembers that it was

Sunday two epistemic moments ago. So, she is sure at m of (SUN at prev2). Therefore,

(30) Cm+(H)=Cs(H on Monday/(MON on Monday)&(W on Monday)&

(SUN at prev0)),

because (MON on Monday)&(W on Monday) is the sequential de-indexicalization of

(MON at prev0)&(W at prev1) under (MON at prev0)&(MON at prev1), and (SUN at prev0)

is the re-indexicalization of (SUN at prev2) for the two epistemic moments earlier time.

Of course, she fully expected on Sunday night that MON would be true on Monday and

that she would wake up on Monday. Also, she knew on that night that it was Sunday. As

a result,

(31) Cm+(H)=Cs(H on Monday)=Cs(H)=1/2.

Therefore, we arrive at the same conclusion whether we apply GSJC to SB’s credal

transition from s to m and then apply it to that from m to m+ step by step, or we apply it

to her credal transition from s to m+ all at once.

This is an intuitive result, and the answer to the given question resembles the

traditional Thirder view of the same question regarding the original SB problem. My next

question is whether we can apply GSJC to the credal transitions in the original SB

problem and acquire the same credence of hers at m+ in H.

131131131131

Here, we are faced with a difficulty. In the original SB problem, it was not

explicitly stated when SB made observations before the experiment began on Sunday

night. Since that information is crucial for using GSJC, we cannot apply that rule to the

original version of the SB problem.

What do we do? We can use GSJC- instead. Here is the rough idea: Perhaps,

there are many possibilities regarding when SB made observations before Sunday night.

However, we can safely assume that those possibilities are irrelevant to how the coin

lands on Monday. Under this assumption, we can apply GSJC- and GSSC

- without

worrying about the mentioned possibilities about what times it had been until Sunday

night.

To precisify this idea, let Rsoo∈O be the partition whose members describe

what days it had been at the epistemic moments before s, where Rso=(D

1o at prev1)&

(D2

o at prev2)&(D3

o at prev3)& … . Let Rmoo∈O be a similar partition such that for

each o∈O, Rmo=(D

1o at prev2)&(D

2o at prev3)&(D

3o at prev4)& … . I make these

assumptions:

(32) For any o∈O, Cs(H/W on Monday)=Cs(H/(W on Monday)&Rso)

and Cs(H/W on Tuesday)=Cs(H/(W on Tuesday)&Rso).

(33) For any o∈O, Cm(H/MON)=Cm(H/MON&Rmo).

54

54 I am assuming that Rs

o is well-specified de priori information with respect to <Cs, H, W on Monday>

and <Cs, H, W on Tuesday>. Similarly, Rmo is well-specified de priori information with respect to <Cm, H,

MON>.

132132132132

In words, from the point of view at s, what days it had been before now is irrelevant to H

conditional on [W on Monday] and, from the point of view at m, what times it had been

before the previous moment is irrelevant to H conditional on MON. These are highly

plausible assumptions. After all, both Rso and Rm

o are tensed propositions describing

what times it had been before Sunday night. Clearly, we have no reason to think that such

a matter is relevant in judging the probability of the coin’s landing heads.

Given these assumptions, I apply GSJC- to SB’s credal transition from s to m:

Consider partition W&MON&(SUN at prev1)&Wm

oo∈O∪W&TUE&(SUN at prev1)&

Wmoo∈O, where Wm

o=(D1

o at prev2)&(D2

o at prev3)&(D3

o at prev4)& … . By definition,

it is SB’s general time-observation partition from s to m.55

Also by definition,

W&MON&(SUN at prev1), W&TUE&(SUN at prev1) is a general time-observation

partition from s to m over [s,m]. Furthermore, the doubleton is sufficiently inclusive.56

Since it is the same partition I used for the credal transition from s to m in SB problem 3,

(27) is also an instance of GSJC- for the credal transition from s to m in the original. By

the same reasoning as in (27)-(29), Cm(H&MON)=1/2Cm(MON), i.e, (29) is also true in

the original SB problem.

55 From SB’s point of view at m, W describes her present observation, (MON at prev0) and (TUE at prev0)

describe the days that are possibly today, and (SUN at prev1)&Wm

o describe what times it had been until the

previous moment.

56

By (32), Cs(H/W on Monday)=Cs(H/(W on Monday)&Rso) and Cs(H/W on Tuesday)=Cs(H/(W on

Tuesday)&Rso) for any o∈O. Since CS(SUN at prev0)=1, Cs(H/(W on Monday)&(SUN at prev0))=Cs(H/(W

on Monday)&(SUN at prev0)&Rso) and Cs(H/(W on Tuesday)&(SUN at prev0))=Cs(H/(W on

Tuesday)&(SUN at prev0)&Rso) for any o∈O. Since (W on Monday) is the re-indexicalization of W under

MON and [(SUN at prev0)&Rso] is the sequential re-indexicalization of ((SUN at prev1)&W

mo) for each

o∈O, W&MON&(SUN at prev1), W&TUE&(SUN at prev1) is sufficiently inclusive, by definition. Done.

133133133133

Next, I apply GSSC- to SB’s credal transition from m to m+. Consider this

partition: MON&(MON at prev1)&(SUN at prev2)&Wm+

oo∈O, where Wm+o=(D

1o at

prev3)&(D2

o at prev4)&(D3

o at prev5)& … . By definition, it is SB’s general time-

observation partition from m to m+.57

Also, think about MON&(MON at prev1)& (SUN

at prev2). By definition, this singleton is her general time-observation partition from m

to m+ over [s,m+]. Moreover, the second partition is sufficiently inclusive.58

Therefore,

(25) is an instance of GSSC- for the credal transition from m to m+. By the same

reasoning as from (25) to (26) in SB problem 3, Cm+(H)= Cm(H&MON)/Cm(MON), i.e.,

(26) is also true in the original SB problem. From (26) and (29), it follows that

Cm+(H)=1/2.

Finally, I apply GSSC- to SB’s credal transition from s to m+. Consider this

partition: [(MON at prev0)&(W at prev1)] & [(MON at prev0)&(MON at prev1)] &

Wm+oo∈O. By definition, it is SB’s general time-observation partition from s to m+.

59

Plus, consider [(MON at prev0)&(W at prev1)] & [(MON at prev0)&(MON at prev1)].

By definition, this singleton is a general time-observation partition from s to m+ over

57 From B’s point of view at m+, MON describes her present observation and time, and (MON at prev1)&

(SUN at prev2)&Wm+

o describes what times it had been until the previous moment.

58

By (33), Cm(H/MON)=Cm(H/MON&Rmo). Since Cm(SUN at prev1)=1, Cm(H/(MON on Monday)&(MON

at prev0)&(SUN at prev1))=Cm(H/(MON on Monday)&(MON at prev0)&(SUN at prev1)&Rm

o). Since (MON

on Monday) is the de-indexicalization of MON under MON and (MON at prev0)&(SUN at prev1)&Rm

o is

the sequential re-indexicalization of (MON at prev1)&(SUN at prev2)&Wm+

o for the one epistemic moment

earlier time, MON&(MON at prev1)& (SUN at prev2) is sufficiently inclusive. 59

From B’s point of view at m+, [(MON at prev0)&(W at prev1)] describes what observations she has made

since the previous moment, [(MON at prev0)&(W at prev1)] describes what days it has been since the

previous moment, and Wm+o describes what days it has been since the two epistemic moments earlier time

(from which she is updating).

134134134134

[s,m+]. Also, the singleton is sufficiently inclusive.60

Hence, (30) is also an instance of

GSSC-. Consequently, Cm+(H)=1/2.

In summary, we can apply GSJC-, or equivalently GSJC, to SB’s credal

transitions from s to m and from m to m+ under some highly plausible assumptions, and

that application leads to coherent results that comply with the popular Thirder view. This

supports my suspicion that GSJC is the correct rule applicable to an agent’s credal

transition from when she did not know what time it was.

H. The Relation between GSJC and Other Rules

In this section, I discuss the relation between GSJC and other rules discussed in the

earlier chapters. First, I will review RSSJC*, a restricted version of SSJC discussed in the

last chapter. Second, I will present RSSJC, which is directly derivable from GSJC, and

compare it with RSSJC*. Finally, I will discuss how the restricted versions of various

rules in this dissertation are derived from GSJC.

First, remember this rule: Let Eo&Voo∈O be B’s sequential time-observation

partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k) and Vo=&1≤k≤m(V

ko at prevm-k).


60 By (32), Cs(H/W on Monday)=Cs(H/(W on Monday)&Rs

o) for any o∈O. Since CS((MON on

Monday)&(SUN at prev0))=1, Cs(H/(MON on Monday)&(W on Monday)&(SUN at prev0))=Cs(H/(MON on

Monday)&(W on Monday)&(SUN at prev0)&Rso). By definition, (MON on Monday)&(W on Monday) is

the sequential de-indexicalization of (MON at prev0)&(W at prev1) under (MON at prev0)&(W at prev1), and

(SUN at prev0) and ((SUN at prev0)&Rso) are the sequential re-indexicalizations for the two epistemic

moments earlier time. By definition, the above singleton is sufficiently inclusive. Done.

135135135135

(RSSJC*) Cn+m(X)= ∑o∈OCn(X in vm


optimal for X, and B was free from temporal ignorance at tn,

where Do is the sequential de-indexicalization of Eo under Vo for each o∈O. Here,

freedom from temporal ignorance at tm means that at tn B knew what time it was then. It

does not imply that at tn B knew anything about what time it was at the even earlier

moments (which we know to be tn-1, tn-2, etc.). Since this is a restrictive rule, it is

reasonable to expect that RGSJC* turns out to be a sub-principle of the new general rule,

GSJC, i.e., all the instances of RGSJC* turn out to be those of GSJC.

Unfortunately, the proviso of RGSJC* is neither strong enough to guarantee that

its instances are all derivable from GSJC nor weak enough to share all interesting

instances of GSJC. To see this point, second, look at this rule: Let Eo&Voo∈O be B’s

sequential time-observation partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k) and

Vo=&1≤k≤m(Vko at prevm-k). Note that Eo&Voo∈O is also B’s general time-observation

partition from tn to tn+m over [tn+1,tn+m]. Then, for any tensed proposition X,

(RSSJC) Cn+m(X)= ∑o∈OCn(X in vm


optimal and sufficiently inclusive for X,

where Do is the sequential de-indexicalization of Eo under Vo for each o∈O. Clearly, this

rule directly follows from GSJC-, which is equivalent to GSJC.

136136136136

To see the difference between RSSJC* and RSSJC, think about the logical

relation between freedom from temporal ignorance and sufficient inclusion. Clearly,

freedom from temporal ignorance is neither necessary nor sufficient for the given

partition’s being sufficiently inclusive. For Eo&Voo∈O is sufficiently inclusive for X

exactly when (#) Cn(X in vm

o/Do)=Cn(X in vm

o/Do&Ro) for any o∈O, where Do is the

sequential de-indexicalization of Eo under Vo and Ro is the sequential re-indexicalization

of Wo, the complement of Eo&Vo, for the m epistemic moments earlier time. To satisfy

(#), either (i) B must have completely known at tn what times it had been until then or (ii)

her credence distribution at tn must have been such that, for any o∈O, what times it had

been is irrelevant to whether X will true in vm

o conditional on what observations she will

make when. Even if B knew at tn what time it was then, it is not sufficient to satisfy (i),

and even if B did not know what time it was at tn, (#) still can be satisfied if (ii) holds.

Given these facts, I find RSSJC to be more attractive as a restricted version of

SSJC. Similar points hold for the other rules that I have discussed: Let E&V be B’s

general time-observation partition from tn to tn+m over [tn+1,tn+m]. Then, for any tensed

proposition X,

(RSSSC) Cn+m(X)=Cn(X in vm/D) if E&V is logically optimal and sufficiently

inclusive for X,

where D is the sequential de-indexicalization of E under V. Let Eo&Voo∈O be B’s time-

observation partition from tn to tn+1. It is also her general time-observation partition from

tn to tn+1 over [tn+1,tn+1]. Then,

137137137137

(RSJC) Cn+1(X)=∑o∈OCn(X in vo/Eo in vo)Cn+m(Eo&Vo) if Eo&Voo∈O is logically

optimal and sufficiently inclusive for X.

Let E&V be B’s time-observation partition from tn to tn+1. Then,

(RSSC) Cn+1(X)=Cn(X in v/E in v) if E&V is logically optimal and sufficiently

inclusive for X.

These principles provide convenient shortcuts for the cases where the direct application

of GSJC is awkward.

When restricted in this way, I believe that SSJC, SSSC, SJC, and SSC become

more safe and attractive rules. As such, GSJC subsumes the intuitions behind the

attractive instances of these rules.

I. Conclusion

At this point, it will be useful to recapitulate the discussions I have presented thus far in

this dissertation. In Chapter II, I developed an updating rule, SJC, which provides an

elegant solution for the SB problem. In Chapter III, I generalized that rule for the credal

transitions between epistemic moments that are not necessarily contiguous. Unfortunately,

the resulting rule, SSJC, yielded different results depending upon whether we apply it to

SB’s credal transitions step-by-step (from s to m and then from m to m+) or all at once

(from s to m+). In this chapter, I presented a rule free from such inconsistency, at least

regarding the SB problem. If I had suggested modifying SSJC to GSJC only to avoid the

138138138138

inconsistent results, I would not be able to avoid the charge of adhocery; however, I have

avoided this charge by providing independent reasons for such modification.

139139139139

CHAPTER V

SATISFACTION OF DESIDERATA

A. Introduction

In the previous chapters, I discussed a series of updating rules. In each chapter, I

suggested a rule for updating one’s de nunc credences (or the degrees of tensed beliefs).

In each case, however, a problem was discovered. Facing each problem, I responded by

suggesting an enhanced rule, immune to the newly discovered problem. In addition, I

provided an independent reason to favor the enhanced rule, on the basis of Gaifman’s

Expert Principle.

In chapter IV, I presented GSJC as the final product of this process. It supports

some instances of SJC and SSJC, which meet a few special conditions. If we restrict SJC

and SSJC with those conditions as provisos, as I believe that we should, we may consider

them to be subordinate rules of GSJC. As such, GSJC is the most general rule discussed

so far.

This raises a question: Is GSJC the most general rule for de nunc updating, full-

stop? In other words, is GSJC a rule such that it is always rational to update one’s de

nunc credences in accordance with it (hereafter: the Final Rule)?61

Recalling my trial and error in the earlier chapters, I find this question to be hard

to answer with confidence. For how can we rule out the possibility that GSJC also suffers

61 Of course, other conditions will have to be satisfied. For example, I will assume that the agent has a

perfect memory about her own past opinions. In this chapter, I will assume that the given agent satisfies

such basic conditions.

140140140140

from a now unknown but serious problem and that there exists a better rule for updating,

which is immune to that problem and defensible for other reasons? Of course, we cannot.

Nevertheless, GSJC might be the Final Rule. Think about the following

properties, which any acceptable rule for updating will satisfy. First, the Final Rule will

produce only synchronically coherent credence functions that diachronically cohere with

the earlier ones: Suppose that the agent has always updated her de nunc credences in

accordance with the Final Rule. Then, her resulting present credence function will satisfy

the standard axioms of probability, and it will be rationally related to any past credence

distribution of the same agent’s. Second, the Final Rule will be observationally

exhaustive: If the agent has updated her de nunc credences in accordance with the Final

Rule, the present credal judgments resulting from such updates will incorporate the

totality of what she has observed until now.

In this chapter, I argue that GSJC satisfies these two requirements. Hence, GSJC

is not disqualified as a candidate for the Final Rule; at least, it is not disqualified due to

any failure to satisfy those requirements. This may not seem significant, but remember

that I already offered a general argument for GSJC. Plus, it generates plausible instances

for various types of credal transitions. In my opinion, when combined, these facts form

good evidence that GSJC is very close to the Final Rule, if not identical.

B. Background

Before I begin the main discussion, I will clarify several notions and assumptions that I

will use and make in this chapter.

First, I clarify the notions of synchronic and diachronic coherences: To some

extent, these words are self-explanatory: An agent’s credence function at tn is

141141141141

synchronically coherent iff the elements of that function cohere with one another, and it

diachronically coheres with the earlier credence functions iff it is rationally related to the

agent’s credence functions at tn-1, tn-2, and so on (up to t0). However, the real challenge is

to provide substantial criteria for such coherence, within the same credence function and

between different credence functions at different times.

If we put aside a few thorny matters (such as the status of Countable Additivity

as an axiom), it is relatively easy to find the criterion for synchronic coherence: An

agent’s credence function Cn at tn is synchronically coherent iff Cn satisfies Non-

negativity, Normality, and Addivity.62

Regarding de dicto credences, a comparable

criterion will be SC or JC: An agent’s credence function Cn diachronically coheres with

her earlier credence functions iff Cn is related by SC/JC to each of Cn-1, …, C0.

Concerning de nunc credences, a comparable criterion will be, hopefully, GSJC: An

agent’s credence function Cn diachronically coheres with the earlier credence functions

iff Cn is related by GSJC to each of Cn-1, …, C0.

Second, I discuss how to apply the notion of transitivity to a rule for updating.

We all know what transitivity is: For any binary relation R and its field S, R is transitive

iff for any x,y,z∈S, if x is related by R to y and y is related by R to z, then x is related by R

to z. However, it is not so obvious that a rule for updating captures a binary relation.

Typically, such a rule relates more than two things. A rule for updating describes the

relation between two credence functions, but its relata include other entities as well. For

example, SC/JC seems to describe the relation among (i) the agent, (ii) her observations

at the given moment, and (iii) her old and new credence functions. However, we can

62 For simplicity, I shall use “Additivity” to refer to Finite Additivity in this chapter. However, everything

discussed here would apply in the same way if I were to use that word to refer to Countable Additivity.

142142142142

focus upon a fixed agent and her history of experiences. With those other elements fixed,

we can regard SC/JC as (capturing) a binary relation between two credence functions.

(Compare: The ancestor-descendant relation appears to have a family as one of the

related entities, but we can treat it as a binary relation by considering only a fixed family

line.) By using a similar method, we can define transitivity for updating principles: Focus

upon a fixed agent B (having such and such an epistemic history H). Then, an updating

rule R is transitive iff for any credence functions Cn, Cn+m, Cn+m+l of B’s, if Cn+m is related

by R to Cn, and Cn+m+l is related by R to Cn+m, then Cn+m+l is also related by R to Cn.

Third, I discuss the notion of “epistemic kernel rules for updating”: As I

mentioned above, a rule for updating describes a relation connecting (at least) an agent’s

credence functions at different moments. Usually, neither of those credence functions is

assumed to play a special role in that relation. For example, think about

“Cn+1(X)=Cn(X/E)” as an instance of SC. In the credal transition described here, we can

say that Cn is the source and Cn+1 is the result but, most likely, Cn was the result of the

given agent’s previous credal transition, and Cn+1 will be the source for her next credal

transition. In this sense, Cn and Cn+1 are not so different in their roles. Let’s call this

category of rule “transitional rules for updating.”

Interestingly, Meacham (2008; forthcoming) suggests a different type of rule for

updating, which he calls “epistemic kernel rules for updating.” According to him, an

epistemic kernel rule for updating relates an agent’s ordinary credence function to a

special credence function, which he calls “the kernel.” When compared with ordinary

credence functions, a kernel is supposed to play a special role. For example, here is an

epistemic kernel version of SC: Cn(X)=C0(X /TE) for any proposition X, where C0 is the

143143143143

agent’s kernel and TE is the totality of her observations until tn. Note that every instance

of this rule includes C0 as the source and never includes it as the result. In this sense, C0

plays a different role from that of any other credence function of the same agent’s.

Exactly what is this so-called kernel? It is difficult to say, because Meacham

does not provide a very clear definition for that notion. Here is my best shot: Your kernel

is a credence function that you would have if you were stripped of all the data from your

past and present observations. For this reason, if you update your credences in

accordance with an epistemic kernel rule, it will transform your kernel into a new

credence function, incorporating the totality of all your observations whatsoever until

now, not just the totality of your observations after some earlier epistemic moment.

While this is an interesting idea, it comes at a cost. First, one may complain that

an agent’s observations might be so crucial for making any credal judgment that if she

were stripped of them, she would not have any remaining credal opinions. Hence, there is

no guarantee that every agent has a kernel in the above sense. Second, “if she were

stripped of all her observational data” seems to be figurative language. How would we

express the above idea more literally? I am not sure, and I suspect that many people will

feel the same way.

To avoid these problems, I adopt this approach: I will consider a set of special

agents, those who had their first credal opinions. Hence, each of them is guaranteed to

have an initial credence function C0, and C0 was literally formed without the help of any

observational data. By considering only those having these features, we can avoid the two

problems mentioned in the previous paragraph.

144144144144

Of course, this approach has its own cost: If we adopt this approach, our

discussion about epistemic kernel rules will be restricted to the credence functions had by

these special agents. In my opinion, this is a quite acceptable cost. For first, almost every

possible agent must have had an initial credence function at some point in her life, except

those rare agents who have lived for an eternal time with no beginning and, for those not

in this elite group, their initial credences are likely to have been based upon very few

observations. Moreover, second, what we can learn from this potentially narrow range of

agents may include important lessons applicable to a broader range of agents, especially

about a special topic such as the rational rule for updating de nunc credences.

In this section, I have clarified the notions and assumptions that I will use

moving forward. Having done this, I am now ready for the main discussions.

C. Strategy

Earlier, I suggested that if an agent has updated her credences in accordance with the

Final Rule, first, her resulting credence function will be synchronically coherent and it

will diachronically cohere with her earlier credence functions, and second, as a form of

judgment, that credence function will incorporate the totality of her observations until the

present moment.

Why do I think that the Final Rule will satisfy these requirements? It is clear why

a rational rule for updating will produce only synchronically coherent credence functions:

For any single credence function of yours, you want its elements to be rationally related.

The standard axioms of probability are meant to capture exactly this relation (in the

domain of subjective probabilities).

145145145145

It is also clear why a rational rule for updating will produce only a credence

function that diachronically coheres with any earlier credence function: For any two

credence functions of yours, you want one of them to be rationally related to the other.

For even if your credence function is internally coherent at any fixed time, you will be

regarded as an unacceptably whimsical agent unless your credence function at each

moment is rationally related to those at the earlier moments. The standard rules for de

dicto updating, strict and Jeffrey conditionalizations, are meant to capture this relation. In

this dissertation, I am trying to find a similar rule for de nunc updating.

How about the requirement of observational exhaustiveness? Many mainstream

epistemologists will agree that an agent’s judgment needs to be based upon what she has

observed. The following thesis, especially, is popular in the literature: An agent’s belief is

justified iff that belief is supported by her evidence. (Connee and Feldman (2004) offer a

similar thesis). I also find it to be an intuitive claim.

However, we should be careful. First, in the literature, “evidence” has been used

frequently with the connotation that its referent is easily accessible to the given agent

(hereafter: accessibility connotation). Perhaps, we can formulate a similar thesis without

being committed to this connotation or sacrificing the intuitive appeal of the above thesis.

For this formulation, I suggest using “observation” to refer to what plays a similar

justificatory role to that of evidence but does not have the accessibility connotation.

Second, even if an observation E supports a hypothesis H when considered in

separation, the rest of one’s observations may include other information that undercuts or

rebuts E’s support for H (Pollock & Cruz, 1986). Hence, if we judge whether an agent’s

belief is justified by considering only fragments of her observations, it may lead to a

146146146146

disastrous result. Therefore, we should make such a judgment by considering the totality

of the given agent’s observations.

These considerations lead to the following refinement of the above thesis: An

agent’s belief is justified iff that belief is justified by the totality of her observations. Can

we formulate an analogous thesis for probabilism? Yes: An agent’s degree of belief is

justified iff that degree of belief is supported by the totality of her observations. This

thesis is plausible for the same reason as its counterpart in the mainstream epistemology:

Intuitively, a rational agent’s credal judgment needs to be based upon her observations,

and if she makes such a judgment on the basis of only a proper subset of her observations,

the rest of her observations may include what would have resulted in a different credal

judgment if considered in making that judgment. In addition, the above thesis is

compatible with the updating model based on JC, in which the agent is not assumed to

have infallible access to her own observations.63

For these reasons, I believe that any ideal general rule for de nunc updating will

satisfy the two mentioned requirements. Of course, even if a certain rule for de nunc

updating, say, R, satisfies those requirements, it is still possible that R fails to satisfy

some other crucial requirements, of which I am not aware yet. However, R’s satisfaction

of the discussed requirements will certainly provide some reason to suspect that R is the

general rule for de nunc updating.

In the rest of this chapter, I shall proceed in the following order: In Section D, I

will present GSJC and GSR in yet new forms. As formulated in these new forms, they

63 Another merit is its compatibility with justificatory externalism, but this does not look like a significant

merit for the standard probabilism, which already assumes a psychological (read: “inside the skull”)

account of beliefs. However, see Williamson (2002) for an externalist version of probabilism.

147147147147

will be called “GSJC+” and “GSR

+.” In Section E, I will prove that GSR

+ captures a

transitive binary relation between credence functions, a crucial lemma for the rest of the

chapter. In Section F, I will show that GSJC+ satisfies the first requirement: If you have

always updated your credences in accordance with GSJC+, your resulting credence

function will be synchronically coherent, and it will diachronically cohere with your

earlier credence functions. In Section G, I will argue that GSJC+ satisfies the second

requirement: Under the same assumption, we can show that your credence functions

resulting from such updates will incorporate the totality of your past observations. In

Section H, I will point out a difference between GSJC, the earlier formulation of my

general updating rule, and GSJC+, the new formulation of that rule presented in this

chapter. I will suggest a way to fill this gap.

D. GSJC+ and GSR

+

In this section, I present GSJC and GSR again, but this time, in a more technically

rigorous way. After that, I discuss their logical relations.

In presenting GSJC and GSR in new forms, I put two restrictions on them: First,

I put a restriction on how the domain of credence functions is constructed. In the earlier

chapters, I didn’t impose any restriction on it except that it consists of tensed propositions.

However, it is usual to assume algebra made out of propositions, events, or sets as the

domain of credence functions. Since we are dealing with tensed propositions, let Γ be the

class of all tensed propositions, and let Ω be an algebra made out of S, i.e., Ω is a subset

of the power set of Γ closed under conjunction and complementation. Additionally, we

assume that Ω is also closed under [… at τ] and [… in ν] operations, i.e., if X is a

member of Ω, then so are [X at τ] and [X in ν], where “τ” is any term referring to a

148148148148

moment, and “ν” is any term referring to an interval. Next, let ∆ be the set of an agent B’s

credence functions. We assume that for any C∈∆, the domain of C is Ω.

Second, I put a restriction on the time intervals and time-specifying tensed

propositions, which will appear in my new formulations of GSJC and GSR. Consider

partitions Φ and Ψ⊆Ω, which meet the following conditions: (i) Φ consists of temporal

intervals, and Ψ consists of tensed propositions specifying what time it is. (ii) For any

v∈Φ, Ψ includes V or the tensed proposition that it is v now. Conversely, for any V∈Ψ,

Φ includes a minimal interval throughout which V is true. (iii) For any X∈Ω and v∈Φ,

the truth-value of X is invariant within v. (iv) Let X and Y be any genuine propositions

that belong to Ω. Suppose that r=C(X/Y&R), where C∈∆ and R=(W1 at prev0)&(W2 at

prev1)&…&(Wm at prevn) for some <W1

, W2,…, W

n>∈Ψn

. Hence, R is a temporal

description or a tensed proposition thoroughly describing what times it has been until

now. In this case, even if we replace R with a better temporal description R*, still

C(X/Y&R*)=r.64

(To see what this means practically, suppose that r=Cn(X in vm/D&R), where

Cn∈∆, X∈Ω, D=(E1 in v

1)&…&(E

m in v

m) for some <E

1…,E

m>∈Ωm

and <v1…,v

m>∈ Φm

,

and R=(W1 at prev 0)&…&(W

m at prevn+1) for some <W

1,…,W

n+1>∈Ψn+1

. By (iii), the

truth-value of X is invariant within vm, that of E

1 is invariant within v

1, …, and that of E

m

is invariant within vm. By (iv), Cn(X in v

m/D&R) is conditioned upon a well-specified

64 Remember the definition of better de priori information in the last chapter: Let R be (W

1 at prev

0)&…&(Wm at prevn) and R* be (W*

1 at prev 0)&…&(W*

m at prevn). Let w

1,...,w

n be intervals associated

with W1,...,W

n; similarly for w*

1,...,w*

n. Then, R* is a better temporal description than R iff (i) for every

k∈1,...,n,wk⊇w∗k

and (ii) for some k∈1,...,n, wk⊃w∗k

.

149149149149

temporal description. Clearly, this allows us to do away with the provisos of optimality in

the earlier formulations of GSJC and GSR.)

Whenever ∆, Ω, Φ, and Ψ satisfy these conditions, I will say that <∆,Ω,Φ,Ψ> is

a model for B’s (de nunc) credences. Given a model <∆,Ω,Φ,Ψ> for B’s credences, we

are ready to formulate the wanted principles: As before, we let Eo&Vo&Woo∈O be an

agent B’s general time-observation partition from tn to tn+m, where Eo=&1≤k≤m(Eko at

prevm-k), Vo=&1≤k≤m(Vko at prevm-k), and Wo=&1≤k≤n+1(W

ko at prevm+k-1). The difference is

that this time, we assume that, for each o∈O, <E1

o,…Em

o>∈Ωm, <V

1o,…,V

mo>∈Ψm

, and

<W1

o,…,Wn

o>∈Ψn. (Hereafter, I will say that Eo&Vo&Woo∈O is constructed from Ω and

Ψ whenever it meets these conditions.) Let Cn, Cn+m∈∆. Consider any X∈Ω. Then, I

make these claims:

(GSJC+) Cn+m(X)=Σo∈OCn(X in v

mo/Do&Ro)Cn+m(Eo&Vo&Wo).

(GSR+) Cn+m(X/ Eo&Vo&Wo)=Cn(X in v

mo/Do&Ro) for each o∈O.

Here, Do is the sequential de-indexicalization of Eo under Vo, and Ro is the sequential re-

indexicalization of Wo for the m epistemic moments earlier time (i.e., Do= &1≤k≤m(Eko in

vko) and Ro=&1≤k≤n+1(W

ko at prevk-1)).

150150150150

Next, I discuss the logical relation between GSJC+ and GSR

+: Let Cn and Cn+m

be B’s credence functions at tn and tn+m. (Hence, Cn, Cn+m∈∆.) Then, we can prove these

two facts:

(1) If Cn+m is synchronically coherent and related by GSR+ to Cn, then Cn+m is

also related by GSJC+ to Cn.

(2) If Cn is synchronically coherent and Cn+m is related by GSJC+ to Cn, then

Cn+m is also related by GSR+ to Cn.

(See APPENDIX B for the proofs.) It is true that these simply mean that GSJC+ is

equivalent to GSR+ because a rational agent’s credence function always will be

synchronically coherent. But it is one of my goals in this chapter to show that if you

always update in accordance with GSJC+, all your credence functions will be

synchronically coherent. Hence, I do not want to assume the synchronic coherence of B’s

credence functions from the outset.

Still, this means that those principles will turn out to be equivalent, if we

independently prove the synchronical coherence of all B’s credence functions. Assuming

such an independent proof, it will be possible to use GSR+ as a proxy for GSJC

+: We can

show that GSJC+ satisfies a wanted requirement by showing that GSR

+ satisfies it.

E. The Transitivity of GSR+

In this section, I argue that GSR+ captures a transitive binary relation between one’s

credence functions at various moments. For an easier discussion, I will first prove an

151151151151

analogous claim for Rigidity, and then provide a proof for the transitivity of GSR+, by

modifying the first proof.

First, I prove the transitivity of (the de dicto version of) Rigidity: If Rigidity

holds between Cn and Cn+m and between Cn+m and Cn+m+l, then it also holds between Cn

and Cn+m+l. To prove this claim, we first suppose these conditions: Let Eii∈I* be a

partition such that each Ei describes a possible course of B’s observations during

[tn+1,tn+m]. Consider Eii∈I such that I⊆I*, Cn+m(Ei)>0 and Σi∈I Cn+m(Ei)=1. In such a

case, we will call Eii∈I “(B’s) observation partition from tn to tn+m.” Let Fjj∈J be also

B’s observation partition from tn+m to tn+m+l in a similar sense. We suppose that

(3) Cn+m(X/Ei)=Cn(X/Ei) for any proposition X and any i∈I, and

(4) Cn+m+l(X/Fj)=Cn+m(X/Fj) for any proposition X and any j∈J

and show that

(5) Cn+m+l(X/Gk)=Cn (X/Gk) for any proposition X and for any k∈K,

where Gkk∈K is B’s observation partition from tn to tn+m+l.

To show this, we need to know how Gkk∈K is related to Eii∈I and Fjj∈J. I

claim that each Gk should be Ei&Fj for some <i,j>∈Ι×J. For remember that each Ei

describes a possible course of observations during [tn+1,tn+m], and each Fk describes a

possible course of observations during [tn+m+1,tn+m+l]. Since each Gk represents a possible

152152152152

course of observations during [tn+ 1,tn+m+l], Gk=Ei&Fj for some <i,j>∈Ι×J. For

convenience, we define Ek=Ei and Fk=Fj when Gk=Ei&Fj. So it suffices to show

(6) Cn+m+l(X/Ek&Fk)=Cn(X/Ek&Fk) for any proposition X and any k∈K.

This is easy to prove: Let X be any genuine proposition. Then, we can derive the

following facts:

(7) Cn(X/Ek&Fk)=Cn(X&Fk/Ek)/Cn(Fk/Ek),

(8) Cn+m(X&Fk/Ek)/Cn+m(Fk/Ek)=Cn+m(X&Ek/Fk)/Cn+m(Ek /Fk), and

(9) Cn+m+l(X&Ek/Fk)/Cn+m+l(Ek/Fk)=Cn+m+l(X/Ek&Fk).

By (3), Cn(X&Fk/Ek)=Cn(X&Fj/Ei)=Cn+m(X&Fj/Ei)=Cn+m(X&Fk/Ek) and Cn(Fk/Ek)=

Cn(Fj/Ei)=Cn+m(Fj/Ei)=Cn+m(Fk/Ek). By (4), Cn+m(X&Ek/Fk)=Cn+m(X&Ei/Fj)=Cn+m+l(X&

Ei/Fj)=Cn+m+l(X&Ek/Fk) and Cn+m(Ek/Fk)=Cn+m(Ei/Fj)=Cn+m+l(Ei /Fj)=Cn+m+l(Ek/Fk). Done.

Second, I prove GSR+’s transitivity. My proof of this fact will be structurally

similar to that of the transitivity of Rigidity, although more complex: Let

Eo&Vo&Woo∈O be B’s general time-observation partition from tn to tn+m, where

Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and Wo=&1≤k≤n+1 (W

ko at prevm+k-

1). Also, let Fp&Vp&Wpp∈P be B’s general time-observation partition from tn+m to tn+m+l,

where Fp=&1≤k≤l(Fkp at prevl-k), Vp=&1≤k≤l (V

kp at prevl-k), and Wp=&1≤k≤n+m+1(W

kp at

153153153153

prevl+k-1). Both are constructed from Ω and Ψ. Then, we suppose that GSR+ holds

between Cn and Cn+m and between Cn+m and Cn+m+l. In other words,

(10) Cn+m(X/Eo&Vo&Wo)=Cn (X in vm

o/Do&Ro) for any tensed

proposition X and o∈O, and

(11) Cn+m+l(X/Fp&Vp&Wp)=Cn+m (X in vlp/Dp&Rp) for any tensed

proposition X and p∈P,

where Do is the sequential de-indexicalization of Eo under Vo, and Ro is the sequential re-

indexicalization of Wo for the m epistemic moments earlier time, and Dp is the sequential

de-indexicalization of Fp under Vp, and Rp is the sequential re-indexicalization of Wp for

the l epistemic moments earlier time. From these suppositions, we prove that GSR+ holds

between Cn and Cn+m+l. Let Gq&Vq&Wqq∈Q be B’s general time-observation partition

from tn to tn+m+l, where Gq=&1≤k≤m+l(Gkq at prevm+l-k), Vq=&1≤k≤m+l(V

kq at prevm+l-k), and

Wq=&1≤k≤n+1(Wkq at prevm+l+k-1). We want to show that

(12) Cn+m+l(X/Gq&Vq&Wq)=Cn(X in vm+l

q/Dq&Rq) for any tensed

proposition X and q∈Q,

where Dq is the sequential de-indexicalization of Gq under Vq, and Rq is the sequential re-

indexicalization of Wq for the m+l epistemic moments earlier time.

154154154154

As before, the key to the proof is in finding the correct relation of

Gq&Vq&Wqq∈Q to Eo&Vo&Woo∈O and Fp&Vp&Wpp∈P. Here is that relation: For

any q∈Q, there exists <o,p>∈O×P such that

(13) Gq=&1≤k≤m(Eko at prevm+l-k)&&1≤k≤l(F

kp at prevl-k);

(14) Vq=&1≤k≤m(Vko at prevm+l-k)&&1≤k≤l(V

kp at prevl-k);

(15) Wq=&1≤k≤n+1(Wko at prevm+l+k-1) and Rq=&1≤k≤n+1(W

ko at prevk-1)=Ro;

(16) Wp=&1≤k≤m(Vko at prevm+l-k)&&1≤k≤n+1(W

ko at prevm+l+k-1) and

Rp=&1≤k≤m(Vko at prevm-k)&&1≤k≤n+1(W

ko at prevm+k-1)=Vo&Wo.

(See APPENDIX C for a proof.)

Having established these facts, I introduce the following definitions: For any

q∈Q, for the <o,p>∈O×P such that (13)-(16) hold between o, p, and q,

(17) Eq=df&1≤k≤m(Eko at prevm+l-k) and Fq=dfFp=&1≤k≤l(F

kp at prevl-k);

(18) VEq=df&1≤k≤m(Vko at prevm+l-k) and VFq= dfVp=&1≤k≤l(V

kp at prevl-k);

and

(19) DEq=dfDo=&1≤k≤m(Eko in v

ko) and DFq=dfDp=&1≤k≤l(F

kp in v

kp),

155155155155

It follows from (13)-(19) that, for any q∈Q,

(20) Gq=Eq&Fq, Vq= VEq&VFq, and Dq= DEq&DFq; and

(21) Wp=VEq&Wq.

Consider any X∈Ω. By (19), it suffices to show that

(22) Cn+m+l(X/Eq&Fq&VEq&VFq&Wq)=Cn(X in vm+l

q/DEq&DFq&Rq).

To show it, first note that

(23) Eo&Vo is equivalent to Do&Vo, and DEq&VEq is equivalent to

Eq&VEq.65

From the above definitions and facts, it follows that for any X∈Ω,

65 In showing the first equivalence, we are trying to show the equivalence between &1≤k≤m(E

ko at prevm+l-

k)&&1≤k≤m(Vko at prevm+l-k) and &1≤k≤m(E

ko in V

ko)&&1≤k≤m(V

ko at prevm+l-k). Clearly, it suffices to show that,

for any number k∈1,...,m, (Eko at prevm+l-k)&(V

ko at prevm+l-k) is equivalent to (E

ko in V

ko)&(V

ko at prevm+l-

k). To show this, consider any k∈1,...,m. (=>) Assume (Eko at prevm+l-k)&(V

ko at prevm+l-k) and show (E

ko

in Vko)&(V

ko at prevm+l-k). By supposition, it was v

ko at the m+l-k epistemic moments earlier time, and E

ko

was true then. By the construction of Ω and Φ, the truth-value of Eko is invariant within v

ko. Hence, E

ko is

true at any t∈vko; by definition, (E

ko in V

ko) is true. Since the supposition provides the other conjunct, done.

(<=) Assume (Eko in v

ko)&(V

ko at prevm+l-k) and show (E

ko at prevm+l-k)&(V

ko at prevm+l-k). By assumption,

Eko is true at any moment in v

ko and it was v

ko at the m+l-k epistemic moments earlier time. Clearly, it

follows that Eko was true at the m+l-k epistemic moments earlier time; by definition, (E

ko at prevm+l-k) is true.

Done. The second equivalence can be shown in a similar way.

156156156156

(24) Cn(X in vm+l

q/DEq&DFq&Rq)=

Cn((X in vm+l

q)&Dp/Do&Ro)/Cn(Dp/Do&Ro),66

(25) Cn+m((X in vm+l

q)&Dp/Eo&Vo&Wo)/Cn+m(Dp/Eo&Vo&Wo)=

Cn+m((X in vm+l

q)&Do/Dp&Rp)/Cn+m(Do/Dp&Rp),67

and

(26) Cn+m+l((X in vm+l

q)&Do/Fp&Vp&Wp)/Cn+m(Do/Fp&Vp&Wp)=

Cn+m+l(X/Eq&Fq&VEq&VFq&Wq).68

Thus, it suffices to show that

(27) Cn+m((X in vm+l

q)&Dp/Eo&Vo&Wo)=Cn((X in vm+l

q)&Dp/Do&Ro),

(28) Cn+m(Dp/Eo&Vo&Wo)=Cn(Dp/Do&Ro),

(29) Cn+m+l((X in vm+l

q)&Do/Fp&Vp&Wp)=Cn+m((X in vm+l

q)&Do/Dp&

Rp), and

(30) Cn+m+l(Do/Fp&Vp&Wp)=Cn+m(Do/Dp&Rp).

These facts follow from (10) and (11).69

Therefore, GSR+ is transitive.

66 For brevity, abbreviate “X in v

m+lq” into “X*.” Then, Cn(X*/DEq&DFq&Rq)=(by (15) and (18))

Cn(X*/Do&Dp&Ro)=Cn(Z1 &Do&Dp&Ro)/Cn(Do&Dp&Ro)=Cn(Z1&Dp/Do&Ro)/Cn(Dp/Do&Ro). Done.

67

Again, abbreviate “X in vm+l

q” into “X*.” Then, Cn+m(X*&Dp/Eo&Vo&Wo)/Cn+m(Dp/Eo&Vo&Wo)=(by

(22)) Cn+m(X*&Dp/Do&Vo&Wo)/Cn+m(Dp/Do&Vo&Wo)=(by (16)) Cn+m(X*&Dp/Do&Rp)/Cn+m(Dp/Do&Rp)=

Cn+m(X*&Dp&Do&Rp)/Cn+m(Dp& Do&Rp)=Cn+m(X*&Do/Dp&Rp)/Cn+m(Do/Dp&Rp). Done. 68

Again, abbreviate “X in vm+l

q” into “X*.” Then, Cn+m+l(X*&Do/Fp&Vp&Wp)/Cn+m(Do/Fp&Vp&Wp)=

Cn+m+l(X*&Do&Fp&Vp &Wp)/Cn+m(Do&Fp&Vp&Wp)=Cn+m+l(X*/Do&Fp&Vp&Wp)=(by (16)-(18) and (20))

Cn+m+l(X*/DEq&Fq&VEq&VFq&Wq)=(by (22)) Cn+m+l(X*/Eq&Fq&VEq&VFq&Wq). Done. 69

Abbreviate “(X in vm+l

q)&Dp” into “X1”and “(X in vm+l

q)&Do” into “X2.” It follows from (8) and (9) that

Cn+m(X1/Eo&Vo&Wo)=Cn(X1 in vm

q/Do&Ro), Cn+m(Dp/Eo&Vo&Wo )=Cn(Dp in vm

q/Do&Ro), Cn+m+l(X2/Fp&Vp

157157157157

This is an important result. Using it as a lemma, I will argue in the next two

sections that GSR+, when regarded as a rule for de nunc updating, satisfies the two

aforementioned requirements that must be met by any candidate for the Final Rule.

F. Synchronic and Diachronic Coherence

In this section, I argue that GSJC+ satisfies the first requirement for the Final Rule. In

other words, if one’s credences have been updated in accordance with GSJC+, the

resulting credence function is, first, synchronically coherent in itself, and second,

diachronically coherent with any earlier credence function.

For my argument, I assume of an agent B that (i) she always updates her

credences in accordance with GSJC+. Under this assumption, I will show the synchronic

and diachronic coherence of the resulting credence function of B’s. To show this, I

depend upon other assumptions as well: (ii) Her initial credence function was

synchronically coherent. (iii) The credence distribution over her general time-observation

partition is always synchronically coherent. Admittedly, neither is trivial, but both are

arguably fairly weak assumptions.70

(Certainly weaker than the results derived from the

assumptions.)

First, I show that B’s credence function is synchronically coherent at any

epistemic moment. Clearly, it suffices to prove that the synchronic coherence of B’s

&Wp)= Cn+m(X2 in vlp/Dp&Rp), and Cn+m(Do/Fp&Vp& Wp)=Cn+m(Do in v

lp/Dp&Rp). Since X1, Dp, X2, and

Do are all genuine propositions, “in vm

q” or “in vlq” is redundant. Done.

70 I find it especially difficult to explain why one’s credence distribution over the general time-observation

partition ought to be synchronically coherent. However, note that it is equally hard to explain why one’s

credence distribution over the observation partition for JC should be synchronically coherent.

158158158158

credence functions is preserved from tn to tn+1. To prove this, assume that Cn∈∆ is

synchronically coherent, i.e., for any X,Y∈Ω,

(NNn) Cn(X)≥0,

(NORMn) Cn(X)=1 if X is tautological, and

(ADDn) Cn(X ∨Y)=Cn(X)+Cn(Y) if ~(X&Y) is tautological.

From this assumption, we prove that Cn+1∈∆ is synchronically coherent, i.e., for any

X,Y∈Ω,

(NNn+1) Cn+1(X)≥0,

(NORMn+1) Cn+1(X)=1 if X is tautological, and

(ADDn+1) Cn+1(X ∨Y)=Cn+1(X)+Cn+1(Y) if ~(X&Y) is tautological.

To prove these facts, we will use the following theorems derivable from NNn, NORMn,

and ADDn: For any X,Y,Z∈Ω

(CNNn) Cn(X/Y)≥0 if defined,

(CNORMn) Cn(X/Y)=1 if defined and X is tautological, and

(CADDn) Cn(X∨Y/Z)=Cn(X/Z)+Cn(Y/Z)

if defined and ~(X&Y) is tautological.

159159159159

First, I show NNn+1: Let X be any tensed proposition, and let Eo&Vo&Woo∈O be B’s

general time-observation partition from tn to tn+1 (constructed from Ω and Ψ). By (i),

(31) Cn+1(X)=∑o∈OCn(X in v1o/Do&Ro)Cn+1(Eo&Vo&Wo),


indexicalization of Wo for the one epistemic moment earlier time. By (iii) and CNNn, all

terms on the right-hand side are non-negative. Hence, NNn+1 is true. Second, I show

NORMn+1: Consider any X that is tautological. Then, (X in v1

o) is also tautological; for, a

tautology is logically true at every moment in any temporal interval. By CNORMn, Cn(X

in v1o/Do&Ro)=1 for any o∈O. By the definition of the general time-observation partition,

∑o∈OCn+1(Eo&Vo&Wo)=1. Hence,

(32) Cn+1(X)=∑o∈OCn(X in v1

o/Do&Ro)Cn+1(Eo&Vo&Wo)=1.

Third, I show ADDn+1: Let X and Y be any members of Ω. Suppose that ~(X&Y) is

tautological. By (i),

(33) Cn+1(X)=∑o∈OCn(X in v1

o/Do&Ro)Cn+1(Eo&Vo&Wo),

(34) Cn+1(Y)=∑o∈OCn(Y in v1o/Do&Ro)Cn+1(Eo&Vo&Wo), and

(35) Cn+1(X∨Y)=∑o∈OCn((X∨Y) in v1

o/Do&Ro)Cn+1(Eo&Vo&Wo).

160160160160

Clearly, [(X∨Y) in v1

o] is equivalent to [(X in v1

o)∨(Y in v1

o)]. Thus,

(36) Cn+1(X∨Y)=∑o∈OCn((X in v1

o)∨(Y in v1o)/Do&Ro)Cn+1(Eo&Vo&Wo).

Because ~(X&Y) is a tautology, [~(X&Y) in v1

o] is also a tautology. Since [(X in v1

o)& (Y

in v1

o)] clearly entails [(X&Y) in v1

o], [(X in v1o)&(Y in v

1o)] is contradictory; hence, ~[(X

in v1

o)&(Y in v1

o)] is another tautology. By CADDn, Cn((X in v1

o)∨(Y in v1o)/Do&

Ro)=Cn(X in v1

o/Do& Ro)+Cn(Y in v1o/Do& Ro). Therefore,

(37) Cn+1(X∨Y)=

∑o∈OCn((X in v1

o)∨(Y in v1

o)/Do&Ro)Cn+1(Eo&Vo&Wo)=

∑o∈O[Cn(X in v1o/Do&Ro)+Cn(Y in v

1o/Do& Ro)]Cn+1(Eo&Vo&Wo)=

∑o∈OCn(X in v1

o/Do&Ro)Cn+1(Eo&Vo&Wo)+

∑o∈OCn(Y in v1o/Do&Ro)Cn+1(Eo&Vo&Wo)

=Cn+1(X)+Cn+1(Y).

In sum, GSJC+ preserves synchronic coherence from Cn to Cn+1. Note that I did not

depend upon the fact that this is a one-step updating. So GSR+ preserves synchronic

coherence from Cn to Cn+m for any m≥1.

Second, I argue that if B updates her credence functions in accordance with

GSJC+, then each of her credence functions diachronically coheres with the earlier

credence function. For this argument, let us consider any Cn, Cn-m∈∆. I will prove that Cn

161161161161

is related by GSJC+ to Cn-m under (i)-(iii);

71 once this proof is done, it will suffice to

argue that Cn diachronically coheres with Cn-m if Cn is related by GSJC+ to Cn-m.

Here goes the proof: Let Cn∈∆ be B’s credence function at tn. We want to show

that Cn is related by GSJC+ to Cn-m∈∆ for any m≥1. By (i), Cn is related by GSJC

+ to Cn-1,

Cn-1 is related by GSJC+ to Cn-2, …, and C1 is related by GSJC

+ to C0. By the earlier result,

Cn, Cn-1, ..., C0 are all synchronically coherent. By (1) and (2), GSJC+ and GSR

+ are

equivalent for those credence functions. Thus, Cn is related by GSR+ to Cn-1, Cn-1 is

related by GSR+ to Cn-2, …, and C1 is related by GSR

+ to C0. By the transitivity of GSR

+,

Cn is related by GSR+ to Cn-1, Cn-2, …, C0. By the equivalence, Cn is related by GSJC

+ to

Cn-1, Cn-2, …, C0. Done.

Next, consider this conditional claim: If Cn is related by GSJC+ to Cn-m, Cn

diachronically coheres with Cn-m, i.e., Cn is rationally related to Cn-m in the relevant sense.

How can I defend this claim? Remember my defense in the last chapter of GSJC as a

rational updating rule. According to it, if B sets her credences at tn by consulting her own

credal opinion at tn-m, then GSJC is the right way to do so. This is because, provided that

B has made a certain sequence of observations after tn-m but she still considers herself at

tn-m as an expert only lacking those observations, there exists a good argument that (a

principle entailing) GSJC captures the restriction that B’s self at tn-m imposes ? on B’s

credences at tn. If this is correct, this restriction will not only justify GSJC as a rule for

updating from Cn-m to Cn, but it also will justify it as a criterion of diachronic coherence

between Cn and Cn-m. Since the difference between GSJC and GSJC+ is ignorable here,

we have an argument for the wanted claim.

71 Since (i) is the antecedent of the wanted claim, it will be eventually discharged, but (ii) and (iii) will

remain to be substantial assumptions.

162162162162

Let me combine my discussions in the above two paragraphs: If an agent has

always updated her credences in accordance with GSJC+ (and (ii)&(iii) are satisfied),

then her present credence function Cn diachronically coheres with any past credence

function Cn-m of hers.

In sum, I have argued that GSJC+ satisfies the first requirement for the Final Rule.

This means that GSJC+ is not ruled out from being the general rule for de nunc updating,

at least not due to any failure to satisfy this first requirement.

G. Observational Exhaustiveness

In this section, I argue that first, an epistemic kernel version of GSJC+ prescribes an ideal

way to incorporate the totality of one’s observations into the present credal judgments,

and second, the original, transitional version of GSJC+ provides a good way to set one’s

credences in accordance with its epistemic kernel counterpart.

To begin, I formulate the epistemic kernel version of GSJC+: I assume that an

agent B had an initial credence function C0 at t0, a moment before she made any

observations. Hence, C0 can play the role of B’s kernel, as required for the formulation of

an epistemic kernel rule. Next, let Eo&Vo&Woo∈Ο be B’s general time-observation

partition from t0 to tn (constructed from Ω and Ψ), where Eo=&1≤k≤n(Eko at prevn-k),

Vo=&1≤k≤n(Vko at prevn-k), and Wo=(Wo at prevn). Given this partition, we can formulate

this rule for updating: For any tensed proposition X,

(GSJCE+

) Cn(X)=Σo∈ΟC0(X in vn

o/Do&Ro)Cn(Eo&Vo&Wo),

163163163163


indexicalization of Wo for the n epistemic moments earlier time (i.e., Do=&1≤k≤n(Eko in

vko) and Ro=Wo for any k∈1,...,n and any o∈Ο).

Why does GSJCE+

provide a good method for an agent to incorporate the totality

of her observations into her credal judgments? First, let’s think about a case in which the

agent has full knowledge about what observations she has made since t1 and what times it

has been since t0. In other words, she is fully certain at tn that she has observed E1, E

2, …,

En and that it has been w, v

1, v

2, …, v

n in those orders, for some <E

1, E

2, …, E

n>∈Ωn

and

<w, v1, v

2, …, v

n>∈Φn+1

. For such a case, GSJCE+

will provide this sub-principle: For any

tensed proposition X,

(GSSCE+

) Cn(X)=C0(X in vn/W&(E

1 in v

1)&(E

2 in v

2)&...&(E

n in v

n)).

Informally, this simply means that

(GSSCE+

) Cn(X)=

C0(X is true during vn/

it is w now&

E1 is true during v

1&

E2 is true during v

2&

...

&En is true during v

n).

164164164164

Observe two facts here: First, the above equation seems to capture a very

plausible way to incorporate the agent’s observations into her credal judgment at tn. For

what could be a more natural way to judge how probable X is at the present moment

(which the agent knows to be in vn) than to judge the probability of “X is true in v

n”

conditional on the conjoined initial truth of “it is w now,” “E1 is true during v

1,” “E

2 is

true during v2,” … and “E

n is true during v

n,” when the agent presently knows that this

conjunction has been confirmed by her observations so far? Second, and more

importantly, those conditions apparently capture all of the agent’s observations until tn.

Since the agent is assumed here to have observed nothing at t0, E1, E

2, …, and E

n exhaust

everything that she has observed until the present moment, tn.

Of course, one may be worried that the agent might not have a full knowledge

about what she has observed and/or what times it has been. For such cases, GSJCE+

provides a comparably intuitive strategy for epistemic kernel updating: If you do not

know what you have observed and/or what times it has been until now, first figure out

what credence you would assign to the target tensed proposition if you knew those facts,

and next take the weighted average of those credences with the weights being your

present credences in various scenarios about your observations and temporal locations

until now. Since each of the sequences of observations and times comprising those

scenarios exhaust all your observations since t1, I believe that GSJCE+

provides a

balanced way to incorporate the totality of your observations into the present credal

judgments.

165165165165

If so, it is easy to argue for the next main claim of this section. Remember the

earlier result that if an agent B has updated her credences in accordance with GSJC+, then

B’s present credence distribution Cn is related by GSJC+ to Cn-1, Cn-2, …, C0. Hence, Cn is

related by GSJC+ to C0 in the mentioned case. But it means that B’s credences at tn are set

in accordance with GSJCE+

.

This result suggests that if an agent updates her credences in accordance with

GSJC+

in the short run and repeats it, she comes to incorporate the totality of her

observations (since the initial moment) into her credences in the long run. In my opinion,

this is a big merit of GSJC+.

H. Filling the Gap

Remember that I used GSJC in the earlier chapters to solve the problem of Sleeping

Beauty. In this section, I first point out that we cannot do the same with GSJC+ because

of a new restriction on the general time-observation partition, and I then offer a solution

based upon a slightly modified version of GSJC+.

First, think about the following difference between GSJC and GSJC+: While

intervals of any size are allowed in the instances of GSJC, only intervals of a very small

size are allowed in those of GSJC+. For the members of Φ are such small intervals of

time that any better specification of one’s temporal location would be meaningless for

judging the relevance of the agent’s (de-indexicalized) observations to the (de-

indexicalized) target tensed proposition. One immediate consequence is that it is difficult

to apply GSJC+ to the cases in which time is specified in relatively coarse-grained units.

In particular, this means that we cannot solve the Sleeping Beauty problem by using

GSJC+. (Certainly, Monday and Tuesday do not belong to Φ.)

166166166166

To overcome this problem, I suggest yet another variant of GSJC, which I will

call “GSJC0.” The main difference between GSJC

+ and GSJC

0 is that time is specified by

using the members of Φ and Ψ in the former, but it is specified by using the unions or

disjunctions of their members in the latter.

To formulate GSJC0, let <∆,Ω,Φ,Ψ> be a model for an agent B’s credence

functions. Then, I will say that <∆,Ω,Θ,Ξ> is an extension of <∆,Ω,Φ,Ψ> iff it satisfies

the following conditions: (i) If an interval v belongs to Φ, then v also belongs to Θ; if

contiguous intervals v1,..., vn all belong to Φ, then ∪1≤j≤nvj also belongs to Φ; finally, no

other interval belongs to Θ. (ii) Ξ is a superset of Ψ such that if an interval w belongs to

Θ, then W or the tensed proposition that it is w now belongs to Ξ, and no other tensed

proposition belongs to Ξ. (Note that Φ can include intervals such as Monday and Tuesday;

correspondingly, Ξ can contain tensed propositions as those expressed by “it is Monday”

and “it is Tuesday.”)

Next, let Ep&Vp&Wpp∈P be an agent B’s general time-observation partition

from tn to tn+m, where Ep=&1≤k≤m(Ekp at prevm-k), Vp=&1≤k≤m(V

kp at prevm-k), and

Wp=&1≤k≤n+1 (Wkp at prevm+k-1) for each p∈P. But this time, we choose V

1p,...,V

mp and

W1

p,...,Wn+1

p not from Ψ but from Ξ. (In this case, I will say that Ep&Vp&Wpp∈P is

constructed from Ω and Ξ.) The rest of the formulation is similar: For any tensed

proposition X,

167167167167

(GSJC0) Cn+m(X)=Σp∈PCn(X in v

mp/Dp&Rp)Cn+m(Ep&Vp&Wp)

if Ep&Vp&Wpp∈P is optimal for X,

where Dp=&1≤k≤m(Ekp in v

kp) and Rp=&1≤k≤n+1 (W

kp at prevk-1). Note that we need the

explicit proviso of optimality because it is not guaranteed to be satisfied here.

Does GSJC0 meet the two requirements for the Final Rule? To answer this

question, we need to identify the logical relation between GSJC+ and GSJC

0. Let Cn and

Cn+m be B’s credence functions at tn and tn+m. Then, these claims are true:

(38) If Cn+m is related by GSJC0 to Cn, then Cn+m is related by GSJC

+ to

Cn, and

(39) If Cn and Cn+m are synchronically coherent and Cn+m is related by

GSJC0 to Cn, then Cn+m is also related by GSJC

+ to Cn.

(See APPENDIX D for the proofs.)

Once these facts are established, it is easy to argue for GSJC0’s satisfaction of the

two requirements for the Final Rule. First, suppose that the given agent, B, has always

updated her credences in accordance with GSJC0 until tn. Thus, Cn is related by GSJC

0 to

Cn-1, Cn-1 is related by GSJC0 to Cn-2, …, C1 is related by GSJC

0 to C0. By (38), Cn is

related by GSJC+ to Cn-1, Cn-1 is related by GSJC

+ to Cn-2, …, C1 is related by GSJC

+ to

C0. Since GSJC+ generates only synchronically coherent credence functions in this case,

so does GSJC0. Moreover, it was proven that, in this case, Cn is related by GSJC

+ to each

168168168168

of Cn-1, Cn-2, …, C0. By (39) and the already proved synchronic coherence of Cn, …, C1,

Cn is related by GSJC0 to each of Cn-1, Cn-2, …, C0. Therefore, GSJC

0 generates a

synchronically coherent credence function that diachronically coheres with the earlier

credence functions (if GSJC0 itself is used as the criterion of diachronic coherence).

Second, because Cn is related by GSJC0 to C0, Cn incorporates the totality of B’s

observations (if B observed nothing at t0).

Therefore, there exists a principle for updating that satisfies the two requirements

for the Final Rule. Furthermore, we can use that rule, GSJC0, in cases in which time is

only coarsely specified, as in the Sleeping Beauty problem.

I. Conclusion

In this chapter, I have argued that if there exists the Final Rule, or a rule that can always

be used by a rational agent to update her de nunc credences, it will satisfy the two

requirements discussed so far. Since GSJC+ (or, equivalently, GSJC

0) was shown to

satisfy those requirements, we have a promising candidate for the Final Rule.

169169169169

CHAPTER VI

CONCLUSION

A. Summary

So far, I have presented and defended GSJC, a new rule for de nunc updating. It has

various merits, the following being the most notable:

First, GSJC provides a convincing solution for the Sleeping Beauty problem: On

the one hand, waking up on Monday seems to be neutral between the coin’s landing

heads and landing tails. Hence, Cm(H/W& MON)=1/2. On the other hand, waking up on

Tuesday entails the coin’s landing tails. Thus, Cm(H/W&TUE)=0. If so, SB has to assign

the weighted average of ½ and 0 to the coin’s landing heads. These facts suggest that

Cm(H)∈(0,1/2). I find this line of reasoning to be convincing, and GSJC supports it

(under several plausible assumptions).

Second, GSJC applies to a wide range of cases, if not all: In the earlier chapters, I

first argued that SJC is the correct rule for de nunc updating, at least in some situations,

and then generalized the rule and argument together to show that GSJC is the generally

correct rule for de nunc updating. So GSJC applies to an agent’s credal transition from tn

to tn+m for any m≥1. Even if the agent is ignorant of what time it is now or what times it

had been until the time from which she is updating, GSJC applies to that credal transition

without a hitch.

Third, GSJC is coherent in many aspects: If you feed a synchronically coherent

prior credence function to that rule, it spits out another synchronically coherent credence

170170170170

function. Moreover, if you continue to update your credences in accordance with GSJC in

the short runs, then you come to have revised your credences in accordance with GSJC in

the long run. Plus, it allows you to incorporate all of your accumulated observations into

your present credal judgments.

In sum, GSJC provides an intuitive solution for the SB problem, it is applicable

to a wide range of cases, and it is coherent in many important aspects. These facts are

good reasons to accept GSJC as the general rule for de nunc updating.

Although these findings are nice achievements, three important issues are still

waiting for our discussion. The first concerns GSJC’s generality beyond de nunc

credences: As it is now, that rule is silent about how to update your de se credences in

general. The second concerns GSJC’s completeness as a rule for updating: If you update

in accordance with that rule, you need a predetermined credence distribution over your

general time-observation partition. Unfortunately, GSJC is silent about how to acquire

such a distribution. The third issue concerns GSJC’s complexity: Undoubtedly, it is a

highly complex rule. If there exists a simpler rule for de nunc/de se updating with the

same merits, isn’t it reasonable to prefer that simpler rule?

In this chapter, I will discuss these issues briefly, but I will refrain from full-

fledged discussions. My primary goals in writing this dissertation were, first, to present a

promising rule for de nunc updating and, second, to develop an argument to defend its

adequacy as the general rule for such an updating process. These goals are ambitious

enough for the first discussion of any updating rule, and so the more advanced

discussions will have to be saved for later papers or books.

171171171171

In the rest of this chapter, I will proceed in this order: In Section B, I will discuss

how to generalize GSJC for de se credences. In Section C, I will discuss in more detail

how to determine the credence distribution over the general time-observation partition. In

Section D, I will discuss the possibility of a simpler rule for de nunc or de se updating

with the same merits as GSJC’s, offering some reasons to be skeptical of that possibility.

B. Remaining Issue 1: Generalization for De Se Updating

At this point, it is natural to ask this question: “What is the correct rule for de se

credences in general?” In this section, I suggest a slightly modified version of GSJC as an

answer to this question.

Let’s begin. As before, let Eo&Vo&Woo∈O be the agent B’s time-observation

partition from tn to tn+m. This time, however, we allow each Eo&Vo&Wo to include any

centered-propositional letters, not just tensed-propositional ones. Then, for any centered

proposition X,

(GSJCde se) Cn+m(X)=Σ o∈O Cn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo) if

Eo&Vo&Woo∈O is optimal for X, and B is sure at tn+m that Cn was her own credence

function m epistemic moments ago,


indexicalization of Wo for the m epistemic moments earlier time. To avoid confusion, I

will call the original, tensed version of GSJC “GSJCde nunc.”

172172172172

What are the differences between GSJCde nunc and GSJCde se? The first one is that

GSJCde se has centered-propositional letters in the places of tensed-propositional ones in

GSJCde nunc, and the second one is that GSJCde se has an additional proviso that the agent is

presently sure that Cn+m is her own credence function at the time from which she is

updating.

Set aside the second modification for a moment. Then, I have suggested simply

replacing tensed-propositional variables with centered-propositional ones. This

suggestion is attractive because scientists and philosophers tend to be conservative: They

always want to preserve as many elements of an established theory when they try to

generalize it for a broader range of cases. Nevertheless, we need to be careful because

this tendency sometimes misfires. For instance, when David Lewis suggested the de se

version of SC (hereafter: SCde se) as the correct rule for de se updating, he was obviously

being driven by the same kind of conservative tendency. Unfortunately, we know now

that SCde se is untenable.

However, I have some hunch that the expansion from GSJCde nunc to GSJCde se

will not misfire in a similar way. To understand why, note that we can conceive of three

types of beliefs: Those about “what this world is,” those about “what time it is now,” and

those about “who I am.” Without a defense, I assume that all beliefs are reducible to these

three types of beliefs or the combinations thereof.

The traditional theories of de dicto credences took only the first type of beliefs

into consideration. That made things easy. For think about this fact: You do not travel

from this world to that world, so the truth-values of the first type of beliefs do not change

through time. As a result, if you learn de dicto evidence E, then you can set your credence

173173173173

in a proposition X by feeding E directly into your previous conditional credence

function.72

In contrast, my theory of de nunc credence covers the second type of beliefs in

addition to the first. Unfortunately, this addition called for a drastic modification of the

traditional updating rules, which resulted in a much more complex rule, GSJCde nunc. The

reason was simple: As time flows, you travel from this time to that time. So the future

becomes the present, the present becomes the past, and the past becomes the farther past.

This fact demands complex techniques for shifting or translating your observations or

time-specifying propositions into the tensed propositions of the matching truth-values,

such as de-indexicalization and re-indexicalization. For example, if you learn “previously,

it was 9 AM,” you will set your present credence by using your previous credence

conditioned upon “it is 9 AM now.”

Now, we are talking about how to theorize the third type of beliefs, involving

“who I am,” in addition to the first and second types. Luckily, it is unlikely to be difficult

this time. Why? No one changes from this person to that person! So my hunch says that I

won’t need a complex technique for shifting in order to update the degrees of my beliefs

about “who I am.” For instance, if I newly learned (S) “I am the son of Sungki Kim,” I

will set my present credence by using my previous credence conditioned just upon S.

(Analogous to the degrees of your beliefs about who you are, are your friend’s beliefs,

your sister-in-law’s, etc.) If this hunch is correct, then it will be okay simply to replace

72 So Cn+1(X)=Cn(X/E). More generally, if your present experience directly affects your credences in Eis,

then your present credence in X is acquired by feeding Ei into your previous conditional credence function

and taking the weighted average. So Cn+1(X)=Σi∈ICn(X/Ei)Cn+1(Ei).

174174174174

tensed proposition letters with centered proposition ones within GSJC to get the general

rule for de se updating.

Still, we need to be careful. Think about this version of the SB problem: SB

problem 4. On Sunday, SB knows she will go through the following experiment: In the

next moment, a group of experimenters will put her to sleep. Then, they will toss a fair

coin. Case 1: The coin lands heads. In this case, she wakes up on Monday knowing that it

is Monday. Case 2: The coin lands tails. In this case, the experimenters duplicate SB

while she is sleeping. Then, they awaken her on Monday, and she knows that it is

Monday. What is her credence on Monday in the coin’s having landed heads?

In this version of the SB problem, when SB wakes up on Monday, she does not

know whether she is SB or the duplicate (hereafter: DUP). Let W be the centered

proposition expressed by “I am waking up today with the memory of SB’s until Sunday

as the last memory.” Given the analogy between the two versions of the problem, it is

likely that

(40) Cm(H)=

Cs(H/W is true of SB on Monday)Cm(W&I am SB)+

Cs(H/W is true of DUP on Monday)Cm(W&I am DUP)∈(0,1/2).

In this equation, SB’s evidence on Monday, W, is de-indexicalized to “W is true of SB on

Monday” or “W is true of DUP on Monday,” depending upon who she is. Clearly, this

fact contradicts my prior hunch that no shifting technique, such as de-indexicalization, is

175175175175

necessary involving the beliefs about “who I am” because no one changes from this

person to that person. What is happening here?

SB problem 4 shows that, although no one can change who she is, a rational

agent can sometimes be unsure about whether she is updating from her own previous

credence function or somebody else’s. From SB’s point of view on Monday, she is

updating from Cs; so, if she is SB, she is updating from her own previous credence

function, but if she is DUP, she is updating from somebody else’s previous credence

function, namely, SB’s.

Here is the general lesson: Usually, a shifting technique such as de-

indexicalization is unnecessary for de se updating because the agent will know the fact

that she is updating from her own prior credence function. However, there are rare cases

in which a rational agent is unsure of this fact. In such a case, some form of shifting

technique will be necessary involving the matter of “who I am.” The new proviso, “B is

sure at tn+m that Cn was her own credence function…” prevents GSJCde se from

erroneously applying to such cases.

In summary, it is acceptable in most of the targeted cases to generalize GSJCde

nunc for de se credences simply by replacing tensed-propositional letters with centered-

propositional ones, but a more complex method for updating will be necessary in some

rare cases. The proviso of GSJCde se is a safety device preventing its misapplication to

such cases.

176176176176

C. Remaining Issue 2: Credence Distribution over the Partition

At this point, a brilliant reader already will be asking this question: “How does an agent

determine her credence distribution over the general time-observation partition?” In this

section, I provide several possible answers.

To begin, I explain what the problem is. For comparison, think about the de dicto

version of JC (hereafter: JCde dicto). According to JCde dicto, an agent’s present credence in

proposition X ought to be the weighted average of her previous credences in X given Ei,

where the weights, the agent’s present credences in Eis, are somehow “directly affected

by” her present observations (Field 1978, p.361; Garber 1980, p.142).

This picture of “probability kinematics” indicates the existence of some relation

between the fineness of the observation partition and the agent’s perceptual power. For

example, suppose that you are watching a piece of cloth under a dim light and it appears

to be red or green to you, but you are unsure which color it is. So your observation

partition is R, G, where R is “this piece of cloth is red” and G is “this piece of cloth is

green.” Now you can imagine that if you had better eyesight, then you could distinguish

subtler colors, such as pinkish red, yellowish red, yellowish green, and bluish green. In

that case, your observation partition would be PR, YR, YG, BG (where PR is “this piece

of cloth is pinkish red,” etc.). Observe that this hypothetical observation partition is more

fine-grained than your actual observation partition. In general, the stronger your

perceptual power is, the finer-grained your observation partition is.

In this sense, your perceptual power sets the limit of the fineness of your

observation partition. This fact suggests that, if there exists a more fine-grained partition

than your current observation partition, then your credences in the members of that

177177177177

partition won’t be directly determined by your observations alone. For that partition will

be too fine-grained for its members’ credences to be determined only by your

observations.

If your perceptual power sets the limit of your de dicto observation partition’s

fineness in this sense, it is reasonable to think that the same is true of your de nunc

observation partition. Unfortunately, this means that at any moment, your credences in

your time-observation propositions cannot be directly determined by your observations.

For simplicity, let’s focus on SJC. According to that rule, Cn+1(X)=Cn(X in vj/Ei in vj)

Cn+1(Ei&Vj), where Cn and Cn+1 are your previous and present credence functions. To

calculate this value, you need to have a credence distribution over Ei&Vj<i,j>∈K at hand.

However, your perceptual power at tn+1 might be incapable of producing such a credence

distribution, if Ei&Vj<i,j>∈K is more fine-grained than Eii∈I.

So we have a problem: Although SJC requires that you have a credence

distribution over your time-observation partition at hand, the partition might be too fine-

grained for your perceptual experience alone to set the credence distribution over it. This

point, of course, generalizes to GSJC; for your general time-observation partition

normally will be more fine-grained than your observation partition.

How can we solve this problem? I do not have a fully developed solution yet, but

I have been considering three possible solutions. Let me outline them one by one. (Again

for simplicity, I will focus upon SJC only, but most of my points below will apply to

GSJC as well.)

First, your credence distribution over the time-observation partition may be

determined by appealing to a principle of indifference. I already discussed this approach

178178178178

in Chapter II, in which I criticized a principle of indifference endorsed by Adam Elga. In

a nutshell, his principle of indifference does not always support a particular credence

distribution over your time-observation partition (it does not if a member of the partition

spans over an uncountable number of worlds), nor is it guaranteed to be consistent (it

leads to a contradiction if the partition includes a possible world with an infinite but

countable number of subjectively indistinguishable centered worlds).

Nevertheless, it is still possible that (i) there exists a new version of the principle

of indifference, (ii) it provides a general recipe for setting your credence distribution over

your time-observation partition, and (iii) it is free from the problem discussed in the last

paragraph. If such a principle is found, it will produce a unique credence distribution over

your time-observation partition. Once such a credence distribution is provided, SJC can

do the rest of the job to calculate your credence function over the whole domain.

Sadly, various paradoxes tainted the reputation of the principle of indifference.

Here is the common structure of those paradoxes: The principle of indifference applies to

a partition, giving the same credence to its members, but in some situations, the same

possibilities are divisible into different partitions, leading to conflicting credence

assignments. Because of this type of problem, many philosophers have rejected the

principle of indifference. Recently, some philosophers have tried to revive it by

formulating a new principle of indifference invulnerable to the mentioned paradoxes.

(For instance, see Elga (2004), Mikkelson(2004), and White (ms.).) I personally am

skeptical of these attempts, but, to be fair, I say that it is too early to make the final

judgment. Of course, whether a principle of indifference can solve our problem depends

upon the success/failure of this general project.

179179179179

Second, if you cannot choose the right credence function over your time-

observation partition, perhaps it will be best not to choose only one of them. Instead, you

might rather have all possible credence distributions compatible with your observations

and previous credence function. Many philosophers suggest that, after all, it is unrealistic

to represent a human opinion with a single credence function because it is humanly

impossible to have such a precise opinion about anything. Instead, they claim that it is

better to represent a human opinion at a certain moment with a set of credence functions

(hereafter: representor) (Sturgeon, 2008; van Fraassen, 1990; Walley, 1991). Accordingly,

each proposition in the domain is assigned a set of real numbers (hereafter: vague

credence). Some advocates of vague credence think that an agent’s vague credence in a

proposition ought to include all compatible values of her credence in that proposition

with her data. I suspect that this idea provides a potential solution to our problem.

To see how this idea works, think about SB’s credal transition from s to m.

Remember that s is SB’s last conscious moment on Sunday, and m is the moment of her

waking up on Monday. Let CFSs be her representor at s. For any Cs∈CFSs, Cs(H)=1/2

because she knows on Sunday that the coin is fair. Let CFSm be her representor at m. At

m, her time-observation partition is W&MON,W&TUE. Since her observations cannot

uniquely determine the credences in W&MON and W&TUE, her vague credence at m in

W&MON ranges over (0,1).73

By SJC, Cm(H)=Cs(H on Monday/W on Monday)

73 Why not [0,1], (0,1] or [0,1)? I tend to think it is crazy that (*) CFSm includes a function which

completely rules out W&MON or W&TUE, since SB lacks any evidence logically contradicting either of

them. However, I admit that I do not have a ready answer to this question: “If (*) is crazy, is it not also

insane that CFSm includes a function that assigns 0.99999… to W&MON or W&TUE, given that she lacks

any evidence supporting either of them to a comparable degree?”

180180180180

Cm(W&MON) + Cs(H on Tuesday/ W on Tuesday)Cm(W&TUE)=1/2Cm(W&MON) for

any Cs∈CFSs and Cm∈CFSm. So her credence value set at m for H ranges over (0,1/2).

In my opinion, this solution is more attractive than the previous one (based on

the principle of indifference) for several reasons. First, as the name suggests, the theory

of vague credence is nothing more than one application of David Lewis’s general theory

of vagueness. If anybody is attracted to his theory of vagueness, she also will be attracted

to the notion of vague credence. Second, the idea of vague credence is motivated by

independent considerations. In reality, nobody can tell what single real number she

assigns to a proposition as the credence. Hence, if we adopt the notion of vague credence,

it will help to build a more realistic model of human credal opinions.

Third, if you are a die-hard subjectivist, you may wonder why there must be a

uniquely rational credence distribution, or even a set of credence distributions, over your

time-observation partition. To understand this point, think about these facts: In the

subjectivist tradition, a rational agent’s credal opinion is not assumed to supervene upon

her accumulated observational data; to establish supervenience, the agent’s initial

credence function needs to be included in the supervenience base. In other words, even if

perfectly rational agents A and B observe exactly the same data throughout their entire

lives, it is possible that A and B will have different credence functions at any time as long

as their initial credence functions were different.

How is it possible that A and B’s initial credence functions were different? Of

course, their initial credal judgments were made before any of their observations. So

there could not have been any a posteriori constraints on their initial credence functions.

Certainly, there must be some a priori constraints, but they will not be sufficient to make

181181181181

them have the same initial credence function. Therefore, this policy looks inevitable:

Allow A to have any initial credence function that she would like as long as it is

synchronically coherent and it satisfies other rational constraints. Mutatis mutandis for B.

Perhaps we can say the same thing about an agent B’s credence distribution over

her time-observation partition. As already emphasized, it is impossible to fully determine

this credence distribution if her time-observation partition is more fine-grained than her

observation partition. In such a case, perhaps this policy will be unavoidable: Allow B to

have any credence distribution over her time-observation partition that she would like as

long as it is synchronically coherent, it satisfies other a priori constraints, and it is

compatible with her credence distribution over her observation partition. Provided a

similar policy for the initial credence function, this laissez-faire policy is not so

implausible any longer.

So far, I have pointed out a problem regarding how to set your credence

distribution over your time-observation partition, necessary for using SJC, and I have

outlined three possible solutions to the problem. I do not pretend that these outlined

solutions are exhaustive; indeed, I do not find them to be fully satisfactory, and I

welcome any new solutions to this problem. If, however, we fail to find a better solution,

we can at least return to those outlined here as our fall-back positions. Of course, all these

problem and solutions are transferrable to GSJC and general time-observation partition.

D. Remaining Issue 3: The Possibility of a Rival Rule

Admittedly, GSJC is a complex rule. We all hate complex rules; they are hard to

understand and difficult to apply to real cases. We would of course prefer a simpler rule

for de nunc updating, all else being equal. However, in this section I argue that a rival

182182182182

rule satisfying the two conditions—“simpler” and “all else being equal”—will be hard to

find, if it exists at all.

The elements making GSJC a complex rule were introduced for some reasons,

after all. A GSJC-er will de-indexicalize her observations, which allows her to avoid the

problem of outdated conditional credence. (Review Chapters II and III.) A GSJC-er will

use the sequential de-indexicalization technique to deal with a sequence of observations,

which allows her to update from a previous, incorrect credence function. (Review

Chapter III.) Finally, a GSJC-er re-indexicalizes any de priori information she has, which

helps to overcome the problem of impoverished temporal knowledge. (Review Chapters

III and IV.) Clearly, these merits come with the cost of additional complexity. Still, the

merits exceed the cost.

Of course, if a simpler rule for de nunc updating enjoys all these merits, we will

favor such a rival rule over GSJC. I am, however, skeptical of this possibility. After all,

such a rival rule also will have to confront the problems mentioned in the last paragraph.

As I’ve shown, these problems can be solved if simplicity is abandoned, but the resulting

modification will not be much superior to GSJC in terms of simplicity. In sum, I am

concerned that a rival rule either will become equally complex after the necessary

modifications are made or will be unable to deal with some of the aforementioned

problems.

To illustrate this dilemma, I will discuss an alternative rule for de se updating as

a case study. Wolfgang Schwarz (ms.) suggests the following rule for de se updating: For

any centered proposition X, define >X to be that X will be true at the next epistemic

moment. For instance, if S is the centered proposition expressed by “I am watching the

183183183183

sun rise,” >S will be the centered proposition that S will be true at the next epistemic

moment. Using this notation, Schwarz formulates a very simple rule for de se updating:

Let X be any centered proposition, and let E be the totality of an agent B’s observations at

tn+1. Then,

(Shifted Conditioning) Cn+1(X)=Cn(>X/>E),

where Cn and Cn+1 are B’s credence functions at tn and tn+1. In words, an agent’s present

credence in a centered proposition X is equal to her previous credence in X’s truth at the

next epistemic moment, given E’s truth at the next epistemic moment, where E is the

totality of her present observations.

Clearly, Shifted Conditioning is a simpler rule than GSJC. However, can a

Shifted Conditionalizer deal with the aforementioned problems, which a GSJC-er can

handle easily? Not all of them.

First, let’s focus on the outdated conditional credence problem. Remember, this

problem: If you use Strict Conditionalization to set your present credence in the present

truth of X, then you come to set it by checking your previous conditional credence in the

previous truth of X.74

So, although you are trying to make a credal judgment of whether X

is presently true, you are doing so by using your previous credal judgment about whether

X was previously true, where the previous and present moments are different. If X is

irreducibly centered, this might be a problem because X’s truth-value might have changed.

74 For brevity, I omit the probabilistic antecedent of the previous conditional credence used for conditioning,

which is your present total evidence E.

184184184184

At first sight, Shifted Conditioning appears to be free from this problem. For, if

you update in accordance with Shifted Conditioning, you come to set your present

credence in the present truth of X by checking your previous conditional credence in the

truth at the then next moment of X.75

In the last sentence, “present” and “next” are co-

referential. As a result, you come to make a credal judgment about whether X is presently

true by using your previous credal judgment about whether X would be true at the then

next moment, just as it should be.

A problem occurs when the agent does not know that the present moment is next

to the moment from which she is updating. For example, when SB wakes up on Monday,

what will her credence that it is Monday be? According to Shifted Conditioning,

Cm(MON)=Cs(>MON/>W)=1.

In other words, her credence on Monday in its being Monday is equal to her credence on

Sunday night that the next epistemic moment would be on Monday, given that she would

wake up (with the memory of Sunday as the last memory) at the next moment. Since she

was sure on Sunday that it would be Monday in the next moment, the above instance of

Shifted Conditioning implies that waking up on Monday, she certainly knows that it is

Monday! This is crazy, because she is not in a position to know that it is Monday.

(One obvious escape route is to assume that actually there are two epistemic

moments next to the one on Sunday night—one on Monday and the other on Tuesday.

75 As before, I omit the probabilistic antecedent of the previous conditional credence used for Shifted

Conditioning, which is >E.

185185185185

For some curious reason, Schwarz rejects this potential solution.76

In any case, Shifted

Conditioning will have to be greatly modified in order to incorporate such a non-linear

structure of nextness among epistemic moments. And that modification will be done only

at the cost of increased complexity.)

What is the origin of this problem? On Sunday night, SB knew that the next

epistemic moment would be on Monday, whether the coin landed heads or tails. However,

when she actually wakes up on Monday, she cannot rule out that it is Tuesday. From her

point of view on Monday, if it is Tuesday, the present time is not next to the epistemic

moment on Sunday night. So she does not know whether or not the present moment is

next to the time from which she is updating. Still, Shifted Conditioning forces her to

update her credence in MON as if she knows that the present moment is next to the time

from which she is updating.

In a nutshell, updating in accordance with Shifted Conditioning can be a mistake

if the agent is not sure about how her present time is related to the time from which she is

updating. So Shifted Conditioning fails to solve the problem of outdated conditioning

credence adequately.

Second, Shifted Conditioning is not as versatile as GSJC because it is meant only

for the credal transition from the agent’s credence function at the previous epistemic

moment. Schwarz seems to be aware of this problem and provides a clue as to how to

remove this limitation: For any centered proposition X, he defines >nX to be the centered

proposition that X will be true at the n epistemic moments later time. For example,

suppose that going to bed, you fully expect to be awakened briefly during the night and to

76 Schwarz’s paper is unclear about this point. In fact, he clarified this point in a private email to me.

186186186186

be awakened by an alarm in the morning. Let WA be the centered proposition expressed

by “I am being awakened by the alarm.” Then, >2WA will be the centered proposition that

WA will be true at the two epistemic moments later time, which you presently know will

be the moment of waking up in the morning tomorrow. Using this expanded notation, I

formulate what I think to be a natural expansion of Shifted Conditioning: Let X be any

centered proposition, and let E1,…,E

m be such that for any k∈1,…,m, E

k is the totality

of her observations at tn+k. Then,

(Sequential Shifted Conditioning) Cn+m(X)=Cn(>mX/&1≤k≤m >kEk),

where Cn and Cn+m are B’s credence functions at tn and tn+m. According to this rule, if an

agent has observed E1, E

2,…, E

m, an agent’s present credence in X is equal to her

conditional credence at the m epistemic moments earlier time in [X’s truth at the m

epistemic moments earlier time], given [the conjunction of the truth at the k epistemic

moments later time of Ek for all k∈1,…,m]. I suppose this is the best way for Schwarz

to go. However, he does not explicitly endorse this extension of Shifted Conditioning,

and, even if he did, the cost would be increased complexity. (Also, remember that this

modification does not solve the outdated conditional credence problem adequately.)

Thus, as it is now, Shifted Conditioning cannot handle some of the problematic

cases with which GSJC has no problem. If you abandon its current simplicity, the thus-

modified rule may do better, but Schwarz will have to pay the cost of increased

complexity. Of course, I cannot completely rule out the possibility that somebody may

find a simpler rule for de nunc/de se updating, which somehow does not suffer from such

187187187187

problems. However, until I actually find such an alternative rule, I will remain skeptical

of that possibility.

E. Conclusion

In this dissertation, I have presented a series of rules for de nunc updating. I discussed

each of them critically, and, whenever I found a problem, I replaced an earlier rule with a

more general and plausible rule for updating. GSJC was the final product of this process.

I defended it by appealing to a variant of Gaifman’s expert principle, and I showed that it

has several highly desirable properties.

In addition, in this chapter I suggested that GSJC might be further generalized to

a rule for de se updating and complemented by some strategy for setting the credence

distribution over an agent’s general time-observation partition. Also, I explained why I

am skeptical about the possibility of any simpler rule for de nunc/de se updating enjoying

all the merits of GSJC. Given all these results and educated hunches, I believe that GSJC

is close to being the general rule for de se updating.

If this belief is correct, what will its general ramifications be? Traditionally, the

following elements have been considered to be the main elements of the theory of

subjective probability:

a) Non-negativity, Normality, and Additivity

b) Strict or Jeffrey-style conditionalization

c) Reflection Principle

d) Principal Principle

188188188188

These principles have been formulated in terms of de dicto credences. The important

question now is, “What happens if we take de se credences into consideration?”

Here is my conjecture: The entire theory of subjective probability needs

substantial modifications for the proper treatment of de se/de nunc credences, except the

synchronic axioms. Let me outline those modifications: In this dissertation, I have argued

that we need to replace SC/JC with GSJC. In his recent paper (2007), Adam Elga offered

a new variant of the Reflection Principle for agents who have lost track of what time it is.

I agree with his view that the original Reflection Principle needs to be modified to deal

with such cases, but I suspect that Elga’s suggestion for that modification is inadequate.

Plus, I believe that we need a new variant of the Principal Principle, which will connect

objective chance and de se credences. To my knowledge, no one previously has

mentioned the need to modify the Principal Principle in order to deal properly with de se

credences.

The theory of de se subjective probability is a vast territory consisting of

uncharted regions. In this dissertation, I explored one of its toughest parts, but the fun is

not over yet. There are still unexplored lands waiting for us.

189189189189

APPENDIX A

EQUIVALENCE BETWEEN GSJC- AND GSJC.

(=>) Suppose GSJC- and show GSJC. Let Eo&Vo&Woo∈O be an agent B’s general

time-observation partition from tn to tn+m, where Eo=&1≤k≤m(Eko at prevm-k),

Vo=&1≤k≤m(Vko at prevm-k), and Wo=&1≤k≤n+1(W

ko at prevm+k-1). By definition,

Eo&Vo&Woo∈O is B’s general time-observation partition from tn to tn+m over [t0,tn+m].

Let X be an arbitrary tensed proposition. By GSJC-,

(1) Cn+m(X)=Σo∈OCn(X in vm


is optimal and sufficiently inclusive for X,


indexicalization of Wo for the m epistemic moments earlier time. For any o∈O,

Eo&Vo&Wo is its own abbreviation and W*o is the vacuous complement of Eo&Vo&Wo.

Clearly, Cn(X in vm

o/Do&Ro)=Cn(X in vm

o/Do&Ro&R*o), where Do is the sequential de-

indexicalization of Eo under Vo, Ro is the sequential re-indexicalization of Wo for the m

epistemic moments earlier time, and R*o is the vacuous sequential re-indexicalization of

190190190190

W*o for the m epistemic moments earlier time.77

Hence, Eo&Vo&Woo∈O is sufficiently

inclusive for X. Therefore,



is optimal for X.

In other words, GSJC is true. Done.

(<=) Suppose GSJC and show GSJC-. Let Eo&Vo&Woo∈O be the general time-

observation partition from tn to tn+m. Given this partition, let Ep&Vp&Wpp∈P be B’s

general time-observation partition from tn to tn+m over [ti≤n,tn+m] i.e., for all p∈P, Ep=Eo,

Vp=Vo, and Wp=&1≤k≤n-i(Wko at prevm+k-1) for some i∈1,…,n+1. By GSJC,



is optimal for X,


indexicalization of Wo for the m epistemic moments earlier time.

77 Since Wp* is the complement of Eo&Vo&Wo for Eo&Vo&Wo, Wp*=&n+1≤k≤n+1(W

kp in prevm+k-1)=T, where

T is a tautology. By definition, Rp* is the re-indexicalization of Wp*, where Rp*=&n+1≤k≤n+1(Wkp in prevk-1).

Since no k satisfies n+1≤k≤n+1, Rp* is vacuously true, i.e., Rp*=T, where T is a tautology.

191191191191

Assume that Ep&Vp&Wpp∈P is optimal and sufficiently inclusive for X i.e.,

these conditions hold: First, for each p∈P, Cn(X in vm

p/Dp&Rp) is conditioned upon well-

specified temporal information, and the truth-value of X is invariant within vm

p and that of

Ekp is invariant within v

kp for any k∈1,…,m. Second, for each o∈O and p∈P, if

Ep&Vp&Wp is an abbreviation of Eo&Vo&Wo and Wp* is the complement, then Cn(X in

vm

p/ Dp&Rp)=Cn(X in vm

p/Dp&Rp&R*p), where Dp is the sequential de-indexicalization

of Ep under Vp, and Rp and Rp* are the sequential re-indexicalizations of Wp and Wp* for

the m epistemic moments earlier time. Given these assumptions, it suffices to show

(4) Cn+m(X)=Σp∈PCn(X in vm

p/Dp&Rp)Cn+m(Ep&Vp&Wp).

To show this, for each p∈P, let Op be the set of o∈O such that Ep&Vp&Wp is an

abbreviation of Eo&Vo&Wo. Clearly, Opp∈P is a partition of O. By this fact and (3),

(5) Cn+m(X)=Σp∈PΣo∈OpCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo) if

Eo&Vo&Woo∈O is optimal for X.

Consider any p∈P. Then, Ep&Vp&Wp is an abbreviation of Eo&Vo&Wo for each o∈Op.

For each o∈Op, let Wo* be the complement of Ep&Vp&Wp for Eo&Vo&Wo. So for each

o∈Op, Eo&Vo&Wo=Ep&Vp&Wp&Wo*. On the one hand, Ep&Vp&Wp is incompatible

with Eo′&Vo′&Wo′ for any o′∉Op by its construction. Since Eo&Vo&Woo∈O is a partition,

this means that Ep&Vp&Wp entails Eo&Vo&Wo for some o∈Op. So Ep&Vp&Wp entails

192192192192

∨o∈Op Eo&Vo&Wo. On the other hand, Eo&Vo&Wo entails Ep&Vp&Wp as Eo&Vo&Wo=

Ep&Vp&Wp&Wo*. In sum, Ep&Vp&Wp is equivalent to ∨o∈Op Eo&Vo&Wo. Hence,

(6) Cn+m(Ep&Vp&Wp)=Σo∈OpCn+m(Eo&Vo&Wo).

Since Ep&Vp&Wp is an abbreviation of Eo&Vo&Wo and Wo* is the complement, Cn(X in

vm

p/Dp&Rp)=Cn(X in vm

p/Dp&Rp&R*o) by (iv), (where …). Since Ep= Eo, Dp= Do. Since

Wo= Wp&W*o, Ro= Rp&R*o. Hence,

(7) Cn(X in vm

p/ Dp&Rp)=Cn(X in vm

p/ Dp&Rp&R*o)=Cn(X in vm

p/Do&Ro).

Thus,

(8) if Eo&Vo&Woo∈O is optimal for X, Cn+m(X)=

Σp∈PΣo∈OpCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo)= (by (5))

Σp∈PCn(X in vm

p/Dp&Rp)Σo∈OpCn+m(Eo&Vo&Wo)= (by (7))

Σp∈PCn(X in vm

p/Dp&Rp)Cn+m(Ep&Vp&Wp). (by (6))

Remember that for all p∈P, Ep=Eo and Vp=Vo. By this fact and the given assumptions, for

each o∈O, Cn(X in vm

o/Do&Ro) is conditioned upon well-specified temporal information,

193193193193

and the truth-value of X is invariant within vm

o and that of Eko is invariant within v

ko for

any k∈1,…,m. In other words, Eo&Vo&Woo∈O is optimal for X. Done.

194194194194

APPENDIX B

EQUIVALENCE BETWEEN GSR+ AND GSJC

+

Let <∆,Ω,Φ,Ψ> be the model for B’s credences and Eo&Vo&Woo∈O be her general

time-observation partition from tn to tn+m constructed from Ω and Ψ, where Eo=

&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and Wo= &1≤k≤n+1(W

ko at prevm+k-1).

Let Cn\, Cn+m∈∆ be her credence functions at tn and tn+m. Then, these facts are provable:

(9) If Cn+m is synchronically coherent and related by GSR+ to Cn, then

Cn+m is also related by GSJC+ to Cn.

(10) If Cn is synchronically coherent and Cn+m is related by GSJC+ to

Cn, then Cn+m is also related by GSR+ to Cn.

To prove (9), suppose that Cn+m satisfies the standard axioms and that Cn+m(X/Eo&Vo

&Wo)=Cn(X in vm

o/Do&Ro) for each o∈O, where Do= &1≤k≤m(Eko in v

ko) and

Ro=&1≤k≤n+1(Wko at prevk-1). Then,

(11) Cn+m(X)=

Σο∈ΟCn+m(X&Eo&Vo&Wo)= (by Additivity)

195195195195

Σο∈ΟCn+m(X/Eo&Vo&Wo)Cn+m(Eo&Vo&Wo)= (def. of P(-/-))

Σο∈ΟCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo). (by supposition)

Done. Note that I used the assumed synchronic coherence of Cn+m in the second line.

To prove (10), suppose that Cn satisfies the standard axioms and that Cn+m(X)=

Σο∈ΟCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo) for any X∈Ω. So Cn and Cn+m are functions

mapping Ω into R. (Note that we are not assuming yet that Cn+m is synchronically

coherent.) For each o∈O, construct a function Po mapping

Ωο=dfX&Eo&Vo&Wo|X∈Ω&o∈O into R as follows:

(12) Po(X&Eo&Vo&Wo)=dfCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo).

Let T be that 0=0, and F be that 0≠0. Thus, these facts are derivable:

(13) Po(Eo&Vo&Wo)=Po(T&Eo&Vo&Wo)=

Cn(T in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo)=Cn+m(Eo&Vo&Wo).

For (T in vm

o) is a tautology (e.g., “That 0=0 is true in June 2009” is a tautology).78

78 A careful reader may complain that if v

mo does not refer to an existing interval, (T in v

mo) might be false.

For example, “That 0=0 is true in June 2009” might be false if the world, including time itself, ceased to

exist at some time in 2008. If such a case is taken into consideration, then I suggest interpreting “X in ν” as

the tensed proposition that is true if and only if, if ν refers to an existing interval of time, then X is true

throughout ν, and, if ν does not refer to any existing interval of time, then X is a tautology.

196196196196

(14) Po(F)=Po(F&Eo&Vo&Wo)=

Cn(F in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo)=0.

For (F in vm

o) is a contradiction (e.g., “That 0≠0 is true in June 2009” is obviously a

contradiction). By (12) and (13),

(15) Po(X&Eo&Vo&Wo)/Po(Eo&Vo&Wo)=Cn(X in vm

o/Do&Ro).

(Also, remember that Po(Eo&Vo&Wo)=Cn+m(Eo&Vo&Wo)>0 by the definition of the

sequential time-observation partition.) Next, we define function P mapping Ω into R as

follows:

(16) P(X)=dfΣο∈ΟPo(X&Eo&Vo&Wo) for any X∈Ω.

Hence,

(17) P(X&Eo&Vo&Wo)=dfΣο∗∈ΟPo((X&Eo&Vo&Wo)&(Eο∗&Vο∗&Wο∗)).

(18) P(Eo&Vo&Wo)=dfΣο∗∈ΟPo((Eo&Vo&Wo)&(Eο∗&Vο∗&Wο∗)).

197197197197

Since Eo&Vo&Woo∈O is a partition, (X&Eo&Vo&Wo)&(Eο∗&Vο∗&Wο∗) is a

contradiction whenever o≠o∗, for any o, o*∈O. Thus,

(19) P(X&Eo&Vo&Wo)=Po(X&Eo&Vo&Wo). (by (17))

(20) P(Eo&Vo&Wo)=Po(Eo&Vo&Wo). (by (18))

So

(21) P(X&Eo&Vo&Wo)/P(Eo&Vo&Wo)=

Po(X&Eo&Vo&Wo)/Po(Eo&Vo&Wo)= (by (19) and (20))

Cn(X in vm

o/Do&Ro). (by (15))

By (12), (16), and supposition,

(22) P(X)=Σο∈ΟCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo)=Cn+m(X)

for any X∈Ω. Hence, P=Cn+m. By substitution in (21),

(23) Cn+m(X&Eo&Vo&Wo)/Cn+m(Eo&Vo&Wo)=Cn(X in vm

o/Do&Ro).

In other words, GSR+ is true. Done. Note that I depended upon the synchronic coherence

of Cn to prove (13) and (14).

198198198198

APPENDIX C

TRANSLATION BETWEEN TEMPORAL CONTEXTS

Remember that Eo&Vo&Woo∈O is B’s general time-observation partition from tn to tn+m,

Fp&Vp&Wpp∈P is B’s sequential time-observation partition from tn+m to tn+m+l, and

Gq&Vq&Wqq∈Q be B’s sequential time-observation partition from tn to tn+m+l, where

(24) Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k), and

Wo=&1≤k≤n+1 (Wko at prevm+k-1),

(25) Fp=&1≤k≤l(Fkp at prevl-k), Vp=&1≤k≤l (V

kp at prevl-k), and

Wp=&1≤k≤n+m+1(Wkp at prevl+k-1), and

(26) Gq=&1≤k≤m+l(Gkq at prevm+l-k), Vq=&1≤k≤m+l(V

kq at prevm+l-k), and

Wq=&1≤k≤n+1(Wkq at prevm+l+k-1).

I claim that

(27) Gq=&1≤k≤m(Eko at prevm+l-k)&&1≤k≤l(F

kp at prevl-k);

(28) Vq=&1≤k≤m(Vko at prevm+l-k)&&1≤k≤l(V

kp at prevl-k);

(29) Wq=&1≤k≤n+1(Wko at prevm+l+k-1) and Rq=&1≤k≤n+1(W

ko at prevk-1)=

Ro; and

199199199199

(30) Wp=&1≤k≤m(Vko at prevm+l-k)&&1≤k≤n+1(W

ko at prevm+l+k-1) and

Rp=&1≤k≤m(Vko at prevm-k)&&1≤k≤n+1(W

ko at prevm+k-1)=Vo&Wo,

tn+m+l, …,tn+m+l tn+m, …, tn+1 tn, …, t0

Time indexicals

used at tn+m+l

prev0,…,prevl-1 prevl, …, prevm+l-1 prevm+l, …, prevn+m+l

Time indexicals

used at tn+m

prev0, …, prevm-1 prevm, …, prevn+m

Time indexicals

used at tn

prev0, …, prevm

Figure 14: Referents of Indexicalsin Different Temporal Contexts. Each row shows B's

indexical specification of various times, from points of views at different times.

To see why, first, see Figure 14: In this figure, we find temporal moments

tn+m+l, ..., t0, to which we refer by using “tn+m+l,” ..., “t0.” However, the given agent, B,

may not have such non-indexical terms to refer to them with. Even in such a case, B can

still use indexicals terms to refer to those moments. In doing so, B can refer to the same

moment(s) by using different time indexicals at different times: For instance, she can

refer to the same moment tn by using “prevm+l” at tn+m+l, by using “prevm” at tn+m, and by

using “prev0” at tn. This means that she can ascribe the same tensed propositions to the

same moment, by believing different tensed propositions at different moments: For

example, let R be the tensed proposition that it is raining in White House now. Then, B

can ascribe R to tn, by believing [R at prevm+l] at tn+m+l, by believing [R at prevm] at tn+m,

and by believing [R at prev0] at tn.

200200200200

tn+m+l, …,tn+m+1 tn+m, …, tn+1 tn, …, t0

Time indexicals

used at tn+m+l


Observations

from tn+1 to tn+m+l

Gm+lq,…, Gm+1

q Gmq,…, G1

q

Time intervals

from t0 to tn+m+l

Vm+lq,…, Vm+1

q Vmq,…, V1

q W1q,…, Wn+1

q

Figure 15: Indexicals, Observations, and Intervals 1.This figure shows how B can ascribe

various observations and time intervals to the past epistemic moments.

Second, see Figure 15: Consider any Gq&Vq&Wq, (where Gq=&1≤k≤m+l(Gkq at

prevm+l-k), Vq=&1≤k≤m+l(Vkq at prevm+l-k), and Wq=&1≤k≤n+1(W

kq at prevm+l+k-1)).

Because of its construction, if B believes Gq&Vq&Wq at tn+m+l, she is ascribing

<Gm+l

q&Vm+l

q, …, G1q&V

1q,W

1q, …, W

n+1q> to <tn+m+l, ...,tn+1, tn, …, t0> at that moment.

79

Similarly, if she does not fully disbelieve Gq&Vq&Wq at tn+m+l, she is not completely

ruling out the ascription of <Gm+l

q&Vm+l

q, …, G1

q&V1

q,W1

q, …, Wn+1

q> to <tn+m+l, ...,tn+1,

tn, …, t0> at that moment. Since Cn+m+l(Gq&Vq&Wq)> 0, B is not fully ruling out that

ascription at tn+m+l.

Third, see Figure 16: Suppose, for reductio, that she ruled out this ascription of

<Gm

q&Vm

q, …, G1

q&V1

q, W1

q, …, Wn+1

q> to <tn+m, ...,tn+1, tn, …, t0> at tn+m. If so, she will

remember at tn+m+l that she has already ruled it out. In this case, B will rule it out at tn+m+l

79 Of course, I am not assuming that B knows at tn+m+l that the moments to which she is ascribing the tensed

propositions are <tn+m+l, …, tn+1, tn,..., t0>. Since she can use indexicals to refer to those epistemic moments,

the ability to identify those moments in non-indexical ways is unnecessary for such an ascription.

201201201201

as well, which contradicts the last paragraph. By reductio, she did not rule out the

ascription of <Gm

q&Vm

q, …, G1

q&V1q,W

1q, …, W

n+1q> to <tn+m, ...,tn+1, tn, …, t1> at tn+m+l.


Time indexicals

used at tn+m+l


Observations

from tn+1 to tn+m+l

Gm+lq,…, Gm+1

q Gmq,…, G1

q

Time intervals

from t0 to tn+m+l

Vm+lq,…, Vm+1

q Vmq,…, V1

q W1q,…, Wn+1

q

Time indexicals

used at tn+m

prev0, …, prevm-1 prevm, …, prevn+m

Observations

from tn+1 to tn+m

Emo,…, E1

o

Time intervals

from t0 to tn+m

Vmo,…, V1

o W1o,…, Wn+1

o

Figure 16: Indexicals, Observations, and Intervals 2. The dark area covers the indexicals,

observations, and times which have no counterpart in Eo&Vo&Wo.

Now, construct E∗&V∗

&W∗ so that E∗

=&1≤k≤m(Gkq at prevm-k), V

∗=&1≤k≤m(V

kq at prevm-k),

and W∗ =&1≤k≤n+1(W

kq at prevm+k-1). Due to its construction, if B fully disbelieved

E∗&V∗

&W∗ at tn+m, she would’ve ruled out the ascription of <G

mq&V

mq, …, G

1q&V

1q,

W1

q, …, Wn+1

q> to <tn+m, ...,tn+1, tn, …, t1> at that moment. Since she does not rule out that

ascription, B does not fully disbelieve E∗&V∗

&W∗ at tn+m. In other words,

Cn+m(E∗&V∗

&W∗)>0. However, remember that Eo&Vo&Woo∈O is B’s general time-

observation partition from tn to tn+m. By definition, the partition exhausts similarly

constructed tensed propositions whose credences at tn+m are strictly positive. Thus, there

must be o∈O such that E∗&V∗

&W∗=Eo&Vo&Wo. For this o∈O,

202202202202

(31) Gm

q=Em

o, …, G1

q=E1

o,

(32) Vm

q=Vm

o, …, V1q=V

1o, and

(33) W1

q=W1

o, …, Wn+1

q=Wn+1

o.

Fourth, see Figure 17: B does not rule out the ascription of <Gm+l

q&Vm+l

q, …,

Gm+1

q&Vm+1

q,Vm

q, …, V1q, W

1q, …, W

n+1q> to <tn+m+l, ..., t0> at tn+m+l. Now, construct

F∗&V∗&W∗ so that F∗=&1≤k≤l(Gm+k

q at prevl-k), V∗=&1≤k≤l (Vm+k

q at prevl-k), and

W∗=&1≤k≤m(Wkq at prevl+k-1)&&1≤k≤n+1(W

kq at prevm+l+k-1). Because of its construction,

if B fully disbelieves F∗&V∗&W∗ at tn+m+l, she would completely rule out the ascription

of <Gm+l

q&Vm+l

q, …, Gm+1

q&Vm+1

q, Vm

q, …, V1

q,W1

q, …, Wn+1

q> to <tn+m+l, ..., t0>

at that moment, which would contradict the above fact. Hence, Cn+m+l(F∗&V∗&W∗)>0.

However, Fp&Vp&Wpp∈P exhausts the similarly constructed tensed propositions whose

credences at tn+m+l are strictly positive.


Time indexicals

used at tn+m+l


Observations from

tn+1 to tn+m+l

Gm+lq,…, Gm+1

q Gmq,…, G1

q

Time intervals

from t0 to tn+m+l

Vm+lq,…, Vm+1

q Vmq,…, V1

q W1q,…, Wn+1

q

Observations from

tn+m+1 to tn+m+l

Fmp,…, F1

p

Time intervals

from t0 to tn+m+l

Vmp,…, V1

p W1p,…, Wm

p Wm+1p,…, Wn+m+1

p

Figure 17: Indexicals, Observations, and Intervals 3. The dark area covers the

observational data in Gq that have no counterparts in Fp.

203203203203

Hence, there exists p∈P such that F∗&V∗&W∗=Fp&Vp&Wp . For this p∈P,

(34) Gm+l

q=Flp, …, G

m+1q=F

1p,

(35) Vm+l

q=Vlp, …, V

m+1q=V

1p,

(36) Vm

q=W1

p, …, V1

q=Wm

p, and

(37) W1

q=Wm+1

p, …, Wn+1

q=Wn+m+1

p.

Therefore,

(38) Gq=&1≤k≤m+l(Gkq at prevm+l-k)= (by (26))

&1≤k≤m(Gkq at prevm+l-k)&&1≤k≤l(G

m+kq at prevl-k)= (by definition)

&1≤k≤m(Eko at prevm+l-k)&&1≤k≤l(F

kp at prevl-k), (by (31)&(34))

(39) Vq=&1≤k≤m+l(Vkq at prevm+l-k)= (by (26))

&1≤k≤m(Vkq at prevm+l-k)&&1≤k≤l(V

m+kq at prevl-k)= (by definition)

&1≤k≤m(Vko at prevm+l-k)&&1≤k≤l(V

kp at prevl-k), and (by (32)&(35))

(40) Wq=&1≤k≤n+1(Wkq at prevm+l+k-1)= (by (26))

&1≤k≤n+1(Wko at prevm+l+k-1). (by (33))

222204040404

Since Rq is the sequential re-indexicalization of Wq for the m+l epistemic moments

earlier time and Ro is the sequential re-indexicalization of Wo for the m epistemic

moments earlier time,

(41) Rq=&1≤k≤n+1(Wko at prevk-1)=Ro. (by definition)

Also,

(42) Wp=&1≤k≤n+m+1(Wkp at prevl+k-1)= (by (25))

&1≤k≤m(Wkp at prevl+k-1)&&1≤k≤n+1(W

m+kp at prevm+l+k-1)= (by definition)

&1≤k≤m(Vkq at prevm+l-k)&&1≤k≤n+1(W

kp at prevm+l+k-1)= (by (36)&(37))

&1≤k≤m(Vko at prevm+l-k)&&1≤k≤n+1(W

ko at prevm+l+k-1). (by (32)&(33))

Since Rp is the sequential re-indexicalization of Wp for the l epistemic moments earlier

time,

(43) Rp=&1≤k≤n+m+1(Wkp at prevk-1)= (by (25))

&1≤k≤m(Wkp at prevk-1)&&1≤k≤n+1(W

m+kp at prevm+k-1)= (by definition)

205205205205

&1≤k≤m(Vkq at prevm-k)&&1≤k≤n+1(W

kq at prevm+k-1)= (by (36)&(37))

&1≤k≤m(Vko at prevm-k)&&1≤k≤n+1(W

ko at prevm+k-1)= (by (32)&(33))

Vo&Wo. (by (24))

Done.

206206206206

APPENDIX D

EQUIVALENCE BETWEEN GSJC+ AND GSJC

0

Let Cn, Cn+m ∈∆ be B’s credence functions at tn and tn+m. Then, these facts are provable:

(44) If Cn+m is related by GSJC0 to Cn, then Cn+m is related by GSJC

+ to

Cn, and

(45) If Cn and Cn+m are synchronically coherent and Cn+m is related by

GSJC+ to Cn, then Cn+m is also related by GSJC

0 to Cn.

It is relatively easy to prove (44): Let <∆,Ω,Φ,Ψ> be a model for an agent B’s

credences and Eo&Vo&Woo∈O be her general time-observation partition from tn to tn+m

constructed from Ω and Ψ, where Eo=&1≤k≤m(Eko at prevm-k), Vo=&1≤k≤m(V

ko at prevm-k),

and Wo=&1≤k≤n+1(Wko at prevm+k-1). Suppose that Cn+m is related by GSJC

0 to Cn, and

show that Cn+m is related by GSJC+ to Cn. Hence, we want to show that

Cn+m(X)=Σo∈OCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo) for any X∈Ω, where Do=

&1≤k≤m(Eko in v


ko at prevk-1). Since Ψ⊆Ξ, Eo&Vo&Woo∈O was

constructed from Ω and Ξ. By supposition, Cn+m(X)=Σo∈OCn(X in vm

o/Do&Ro)

Cn+m(Eo&Vo&Wo) if (i) the truth-value of X is invariant within vm

o and that of Eko is

207207207207

invariant within vko for each k∈1,...,m and (ii) Cn(X in v

mo/Do&Ro) is conditioned upon

a well-specified temporal information, for each o∈O. Since X∈Ω and Eo&Vo&Woo∈O

was constructed from Ω and Ψ, (i) and (ii) are satisfied. Done.

It is more difficult to prove (45): Let <∆,Ω,Θ,Ξ> be an extension of <∆,Ω,Φ,Ψ>

and Ep&Vp&Wpp∈P be B’s general time-observation partition from tn to tn+m constructed

from Ω and Ξ, where Ep=&1≤k≤m(Ekp at prevm-k), Vp=&1≤k≤m(V

kp at prevm-k), and

Wp=&1≤k≤n+1(Wkp at prevm+k-1). Suppose that Cn and Cn+m are synchronically coherent

and Cn+m is related by GSJC+ to Cn, and show that Cn+m is also related by GSJC

0 to Cn.

To show this, let X be any member of Ω. It suffices to assume the satisfaction of (i) and

(ii) and show that Cn+m(X)=Σp∈PCn(X in vm

p/Dp&Rp) Cn+m( Ep&Vp&Wp), where Dp=

&1≤k≤m(Ekp in v

kp) and Rp=&1≤k≤n+1(W

kp at prevk-1). By construction, there exists

Ψ<k,p>⊆Ψ such that Vkp≡∨Ψ<k,p> for each k∈1,...,m and p∈P and there also exists

Ψ∗<k,p>⊆Ψ such that Wkp≡∨Ψ∗<k,p> for each k∈1,...,n+1 and p∈P. For each p∈P,

construct Eo&Vo&Woo∈O*p such that Eo=Ep, Vo=&1≤k≤m(Vko at prevm-k) for some

<V1

o,...,Vm

o>∈Ψ<1,p>×...×Ψ<m,p>, and Wo=&1≤k≤n+1(Wko at prevm+k-1) for some

<W1

o,...,Wn+1

o>∈Ψ∗<1,p>×...×Ψ∗<n+1,p>. Clearly, Eo&Vo&Woo∈O*p is a partition.

Consider arbitrary p∈P. On the one hand, ∨o∈O*pEo&Vo&Wo entails Ep&Vp&Wp.

To see this fact, it suffices to show that Eo&Vo&Wo entails Ep&Vp&Wp for any o∈O*p.

208208208208

So consider any o∈O*p. First, Eo clearly entails Ep. Second, Vo entails Vp. Why? For each

k∈1,...,m, (Vko at prevm-k) entails (V

ko at prevm-k), because V

ko∈Ψ<k,p> and so V

ko entails

∨Ψ<k,p>, which is the same as Vkp. It clearly follows that &1≤k≤m(V

ko at prevm-k) entails

&1≤k≤m(Vkp at prevm-k). Third, Wo entails Wo. Why? For each k∈1,...,n+1, (W

ko at

prevm+k-1) entails (Wko at prevm+k-1), because W

ko∈Ψ∗<k,p> and so W

ko entails ∨Ψ∗<k,p>,

which is the same as Wkp. It clearly follows that &1≤k≤n+1(W

ko at prevm+k-1) entails

&1≤k≤n+1(Wkp at prevm+k-1).

On the other hand, Ep&Vp&Wp entails ∨o∈O*pEo&Vo&Wo: By the construction of

Eo&Vo&Woo∈Op, there exists some o∈O*p such that Eo=Ep, Vo=&1≤k≤m(Vko at prevm-k)

for some <V1

o,...,Vm

o>∈Ψ<1,p>×...×Ψ<m,p>, and Wo=&1≤k≤n+1(Wko at prevm+k-1) for some

<W1

o,...,Wn+1

o>∈Ψ∗<1,p>×...×Ψ∗<n+1,p>. So Vko entails ∨Ψ<k,p> for each k∈1,...,m.

Similarly, Wko entails ∨Ψ∗<k,p> for each k∈1,...,n+1. Since V

kp≡∨Ψ<k,p> and

Wkp≡∨Ψ∗<k,p>, V

ko entails V

kp for each k∈1,...,m and W

ko entails W

kp for each

k∈1,...,n+1. Clearly, (Vko at prevm-k) entails (V

kp at prevm-k) for each k∈1,...,m and

(Wko at prevm+k-1) entails (W

kp at prevm+k-1) for each k∈1,...,n+1. Thus, Vp&Wp entails

Vo&Wo. Since Ep clearly entails Eo, Ep&Vp&Wp entails that Eo&Vo&Wo for some o∈O*p.

Hence, Ep&Vp&Wp entails ∨o∈O*pEo&Vo&Wo.

209209209209

Since p was arbitrarily chosen from P, Ep&Vp&Wp is equivalent to

∨o∈O*p(Eo&Vo&Wo) for each p∈P. For each p∈P, construct Op⊆O*p such that

Cn+m(Eo&Vo&Wo)>0 for any o∈Op. Define O to be ∪p∈POp. Hence,

(46) Cn+m(Eo&Vo&Wo)>0 for any o∈O,

(47) Σo∈OCn+m(Eo&Vo&Wo)=

Σp∈PΣo∈OpCn+m(Eo&Vo&Wo)= (by the construction of O)

Σp∈PCn+m(∨o∈Οp(Eo&Vo&Wo))= (by Additivity)

Σp∈PCn+m(∨o∈O*p(Eo&Vo&Wo))= (by the construction of each Op)

Σp∈PCn+m(Ep&Vp&Wp)=1, and (by the above equivalence)

(48) Σo∈OCn(Do&Ro)>0. (by (46))

To understand how (48) derives from (46), suppose, for reductio, that Σo∈OCn(Do&Ro)=0.

Then, Cn(Do&Ro)=0 for any o∈O. It means that at tn, B completely rules out the

possibility that [E1

o will be true in v1

o,E2o will be true in v

2o, …, E

mo will be true in v

1o]

and [it is w1o now, it was w

1o at the one epistemic moment earlier time, …, it was w

n+1o at

the n+1 epistemic moment earlier time]. At tn+m, B remembers that she ruled out this

possibility at the m epistemic moments earlier time. Acknowledging that m epistemic

210210210210

moments have passed, she will rule out the possibility that [E1o&V

1o was true, E

2o&V

2o

was true, …, Em

o&Vm

o is true now] and [it was w1o at the m epistemic earlier time, it was

w1o at the m+1 epistemic moment earlier time, …, it was w

n+1o at the m+n+1 epistemic

moment earlier time]. So Cn+m(Eo&Vo&Wo)= 0. However, this contradicts (46). By

reductio, Σo∈OCn(Do&Ro)>0. From (46)-(48), it follows that Eo&Vo&Woo∈O is B’s

sequential time-observation partition from tn to tn+m over [t0,tn+m] constructed from Ω and

Ψ. Since we supposed that Cn+m is related by GSJC+ to Cn,

(49) Cn+m(X)=Σο∈ΟCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo).

Focus upon the value of Cn+m(Eo&Vo&Wo). For any o∈O such that o∈Op,

(50) Cn+m(Eo&Vo&Wo)=

Σp∈PΣo∈OpCn+m(Eo&Vo&Wo)= (by the construction of O)

Σp∈PΣo∈O*pCn+m(Eo&Vo&Wo)= (by the construction of Op)

Σp∈PCn+m(∨o∈O*p(Eo&Vo&Wo))= (by Additivity)

Σp∈PCn+m(Ep&Vp&Wp). (Ep&Vp&Wp≡ ∨o∈O*p(Eo&Vo&Wo))

Focus upon the value of Cn(X in vm

p/Dp&Rp). For any p∈P and o∈Op,

211211211211

(51) Cn(X in vm

p/Dp&Rp)=

Cn(X in vm

p/&1≤k≤m(Ekp in v

kp)&&1≤k≤n+1(W

kp at prevk-1))= (by definition)

Cn(X in vm

o/&1≤k≤m(Ekp in v

kp)&&1≤k≤n+1(W

ko at prevk-1)).

For Cn(X in vm

p/Dp&Rp) was assumed to be conditioned upon well-specified temporal

description and wko⊆w

kp, for each k∈1,...,n+1. For any p∈P and o∈Op,

(52) Cn(X in vm

o/&1≤k≤m(Ekp in v

kp)&&1≤k≤n+1(W

ko at prevk-1))=

Cn(X in vm

p/&1≤k≤m(Eko in v

kp)&&1≤k≤n+1(W

kp at prevk-1))=

Cn(X in vm


ko)&&1≤k≤n+1(W

kp at prevk-1)),

because the truth-value of X is invariant within vm

p and that of Eko (=E

kp) is invariant

within vkp, and v

mo⊆v

mp and v

ko⊆v

kp, for any k∈1,...,m. Hence,

(53) Cn+m(X)=

Σο∈ΟCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo)= (by (49))

Σο∈Οp&p∈PCn(X in vm

o/Do&Ro)Cn+m(Eo&Vo&Wo)= (by construction of O)

Σο∈Οp&p∈PCn(X in vm

p/Dp&Rp)Cn+m(Ep&Vp&Wp)= (by (50)-(52))

212212212212

Σp∈PCn(X in vm

p/Dp&Rp)Cn+m(Ep&Vp&Wp). (simplification)

Therefore, Cn+m is related by GSJC+ to Cn. Done.

213213213213

BIBLIOGRAPHY

Alchourron, C.E., Gardenfors, P., Makinson, D. (1985). On the Logic of Theory Change.

Journal of Symbolic Logic, 50, 510-530.

Bradley, D. (2003). Sleeping Beauty: a note on Dorr's argument for 1/3. Analysis, 63 ,

266-268.

Bricker, P. (ms.) Realism without Parochialism. An unpublished manuscipt.

Connee, E., & Feldman, R. (2004). Evidentialism. Oxford: Oxford University Press.

Dietrich, F., & List, C. (forthcoming). The Aggregation of Propositional Attitudes:

Towards a General Theory. In T. S. Gendler, Oxford Studies in Epistemology.

Oxford: Oxford University Press.

Dorr, C. (2002). Sleeping Beauty: in defence of Elga. Analysis, 62 , 292-296.

_______ (2000). Self-locating Beliefs and Sleeping Beauty Problem. Analyssis, 60 , 143-

147.

_______ (2004). Defeating Dr. Evil with Self-locating Belief. Philosophy and

Phenomenological Research, 69 , 383-396.

Elga, A. (2007). Reflection and Disagreement. Nous, 41 , 478-502.

Field, H. (1978). a Note on Jeffrey Conditionalization. Philosophy of Science, 45 #3 ,

361-367.

Gaifman, H. (1988). A Theory of Higher Order Probabilities. In B. Skyrms, & W. Harper,

Causation, Chance, and Credence (pp. 191-220). Dortrecht: Kluwwer Academic

Publishers.

Garber, D. (1980). Field and Jeffrey Conditionalization. Philosophy of Science, 47 #1 ,

142-145.

Hajek, A. (2003). What Conditional Probability Could Not Be. Synthese, 137 #3 , 273-

323.

Hall, N. (2004). Two Mistakes about Credence and Chance. Australian Journal of

Philosophy, 82 , 93-111.

214214214214

Jeffrey, R. (1984). Baysianism with a Human Face. In J. Earman, Testing Scientific

Theories: Minnesota Studies in the Philosophy of Science (pp. 133-156).

Minneapolis: University of Minnesota Press.

_______ (1990). The Logic of Decision. London: University of Chicago Press.

Katsuno, H., & Mendelzone, A. O. (1992). On the Difference between Updating a

Knowledge Base and Revising it. In P. Gardenfors, Belief Revision (pp. 183-203).

Cambridge: Cambridge University Press.

Kierland, B., & Monton, B. (2005). Minimizing Inaccuracy for Self-Locating Beliefs.

Philosophy and Phenomenological Research, 70, 384-395.

Lewis, D. K. (1979). Attitudes De Dicto and De Se. The Philosophical Review, 88 , 513-

543.

_______ (1986). A Subjectivist's Guide to Objective Chance. In R. Jeffrey, Studies in

Inductive Logic and Probability, Vol. II. (pp. 263-293). Berkeley: University of

Chicago Press.

_______ (2001). Sleeping Beauty: Reply to Elga. Analysis, 61 , 171-176.

Meacham, C. J. (2008). Sleeping Beauty and Dynamics of De Se Beliefs. Philosophical

Studies, 138 , 245-269.

_______ (forthcoming). Unraveling Tangled Web: Continuity, Internalism, Uniqueness

& Self-Locating Belief. In T. S. Gendler, & J. Hawthorne, Oxford Studies in

Epistemology, Vol. 3. Oxford: Oxford University Press.

Mikkelson, M. J. (2004). Dissolving the Wine/Water Paradox. British Journal of

Philosophy of Science, 55, 137-145.

Pollock, J., & Cruz, J. (1986). Contemporary Theories of Knowledge. Lanham: Rowman

& Littlefield.

Prior, A. N. (2003). Papers on Time and Tense. Oxford: Oxford University Press.

Rescher, R., & Urquhart, A. (1971). Temporal Logic. New York: Springer-Verlag.

Schwarz, W. (ms.). Changing Minds in a Changing World. An unpublished manuscript.

Sturgeon, S. (2008). Reason and the Grain of Belief. Nous, 42 #1 , 139-165.

Talbott, W. J. (1991). Two Principles of Bayesian Epistemology. Philosophical Studies,

62 , 135-150.

215215215215

van Fraassen, B. (1984). Belief and the Will. Journal of Philosophy, 81 , 235-256.

_______ (1990). Figures in a Probability Landscape. In A. Gupta, & M. Dunn, Truth or

Consequences (pp. 345-356). Dortrecht: Kluwer.

Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. Chapman & Hall.

Weatherson, B. (2005). Should We Respond to Evil with Indifference? Philosophy and

Phenomenological Research, 70 , 614-635.

Weintraub, R. (2004). Sleeping Beauty: a simple solution. Analysis, 64 , 8-10.

White, R. (ms.) Evidential Symmetry and Mushy Credence. An unpublished manuscript.

Sleeping Beauty and De Nunc Updating

Documents

Transcript of Sleeping Beauty and De Nunc Updating